#  Special flow equation and GKP-Witten relation

Sinya Aoki1, Janos Balog2, Tetsuya Onogi3 and Shuichi Yokoyama1 1 Center for
Gravitational Physics and Quantum Information, Yukawa Institute for
Theoretical Physics, Kyoto University, Kitashirakawa-Oiwakecho, Sakyo-Ku,
Kyoto 606-8502, Japan
2 Institute for Particle and Nuclear Physics, Wigner Research Centre for
Physics, H-1525 Budapest 114, P.O.B. 49, Hungary
3 Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

###### Abstract

We develop a framework for the reconstruction of the bulk theory dual to
conformal field theory (CFT) without any assumption by means of a flow
equation. To this end we investigate a minimal extension of the free flow
equation and find that at a special parametrization the conformal
transformation for a normalized smeared operator exactly becomes the isometry
of anti-de Sitter space (AdS). By employing this special flow equation to
O$(N)$ vector models, we explicitly show that the AdS geometry as well as the
scalar field satisfying the GKP-Witten relation concurrently emerge in this
framework.

††preprint: YITP-22-36
OU-HET-1140

##  I 1\. Introduction

The AdS/CFT duality plays a key role in understanding the holographic aspect
of gravitation and may give a hint for quantum gravity Maldacena:1997re .
Although this duality is supposed to originate in the open/closed string
duality and can be tested by explicit calculations Gubser:1998bc ;
Witten:1998qj , the full understanding of the duality including the mechanism
of the emergence of the additional direction has not been attained yet. (See
Heemskerk:2010hk  for instance.)

While several approaches for the bulk reconstruction from boundary CFT have
already developed considerably (see Aharony:1999ti ; DHoker:2002nbb ;
Skenderis:2002wp ; Benna:2008yg ; Harlow:2018fse ; Kajuri:2020vxf  for reviews
and references therein), they assume the existence of an AdS spacetime a
priori and only succeeded in reproducing excitations around the AdS
background. The present authors have explored an alternative approach
employing a course-graining technique called flow equation Aoki:2015dla ;
Aoki:2016ohw ; Aoki:2016env ; Aoki:2017bru ; Aoki:2017uce ; Aoki:2018dmc ;
Aoki:2020ztd  to realize an emergent bulk AdS geometry itself from CFT. A flow
equation generally describes a non-local course-graining of the elementary
fields of a given quantum field theory so as to remove contact divergences.
This is different from the standard approach, in which the course-graining is
applied to singlet or gauge invariant operators. Due to this property the flow
equation enables one to construct a class of composite operators which behave
as geometric objects in the holographic space after taking the quantum
average. In particular, taking the vacuum expectation value (VEV) of the
metric operator yields a symmetric tensor in a one dimension higher space,
which has the interpretation as a certain information metric Aoki:2017bru .
This is an advantage in this framework, which allows the reconstruction of the
bulk theory without assuming the bulk geometry from the beginning.

The study by means of the flow equation so far has been focused only on the
bulk reconstruction for the vacuum state in CFT. The next step is to extend
this reconstruction to excited states. (See also Kabat:2011rz ; Botta-
Cantcheff:2015sav ; Kabat:2015swa ; Christodoulou:2016nej .) The goal of this
paper is to develop a framework for it. In fact it is possible to continue
this program by using the free flow equation as was done previously, but our
preliminary studies indicated that our results for excited scalar states using
the simplest smearing are difficult to give a geometric interpretation. In
particular, the bulk scalar field constructed using the flow equation does not
satisfy the equations of motion on the AdS background.

In this letter, we propose a modified flow equation and study correlation
functions on the emergent bulk spacetime involving excited states by a primary
scalar operator. We find that some special flow equation maps the conformal
symmetry to the bulk AdS isometry at the operator level and allows us to have
the expected bulk interpretation for both the background geometry and a scalar
field excitation, as we shall see below.

##  II 2\. Framework

###  II.1 2-1. Interpolating flow

In this section we develop a framework to construct the bulk counterpart for a
primary scalar state in Euclidean CFT by the flow equation approach. For a
concrete illustration, we consider a $d$-dimensional $O(N)$ vector model with
conformal symmetry. We denote the elementary scalar field in the vector
representation of $O(N)$ by $\varphi^{a}(x)$. The two-point correlation
function is
$\left\langle\varphi^{a}(x_{1})\varphi^{b}(x_{2})\right\rangle=\delta^{ab}{C_{0}\over|x_{1}-x_{2}|^{2\Delta}}$
where $C_{0}$ is a constant and $\Delta$ is the conformal dimension. In the
previous studies of the flow equation approach, we have used the simplest
smearing defined by the free flow equation

$\displaystyle\partial_{\eta}\phi^{a}(x;\eta)=\partial^{2}\phi^{a}(x;\eta),$
(1)

with $\phi^{a}(x;0)=\varphi^{a}(x)$, where $\eta\geq 0$ is the flow parameter,
and $\partial^{2}=\delta^{\mu\nu}\partial_{\mu}\partial_{\nu}$. However, as
mentioned in the introduction, we have encountered a difficulty in the bulk
interpretation for an excited state by this simplest smearing.

Therefore we shift the gear to modify the flow equation suitably for this
purpose. A minimal extension of the free flow equation (1) is to add a second
derivative term with respect to $\eta$:

$(-\alpha\eta\partial_{\eta}^{2}+\beta\partial_{\eta})\phi^{a}(x;\eta)=\partial^{2}\phi^{a}(x;\eta),$
(2)

where $\alpha$ and $\beta$ are dimensionless real parameters. For later
convenience, we refer to this flow as the interpolating flow. We restrict the
range of parameters as $\alpha,\beta\geq 0$ and it turns out that one can find
a desired flow even with this condition. We solve the equation (2) performing
the convolution $\phi^{a}(x;\eta)=\int d^{d}y\rho_{\eta}(x-y)\varphi^{a}(y)$,
with $\rho_{0}(x)=\delta^{d}(x)$. In the Fourier space the smearing function
$\rho_{\eta}$ satisfies

$(-\alpha\eta\partial_{\eta}^{2}+\beta\partial_{\eta})\rho_{\eta}(p)=-p^{2}\rho_{\eta}(p),$
(3)

where $\rho_{0}(p)=1$. This differential equation can be solved by using the
modified Bessel function of the second kind as
$\rho_{\eta}(p)={2\over\Gamma(\nu)}\widetilde{p}^{\nu}K_{\nu}\left(2\widetilde{p}\right)$,
where $\widetilde{p}=\sqrt{\frac{\eta}{\alpha}}p$ with $p=\sqrt{p^{2}}$ and
$\nu=(\alpha+\beta)/\alpha$. Another independent solution involving
$K_{-\nu}\left(2\widetilde{p}\right)$ is excluded by the initial condition.
Then the smearing function in the coordinate space is computed as

$\displaystyle\rho_{\eta}(x)$ $\displaystyle=$
$\displaystyle\frac{2}{\Gamma(\nu)}{2\pi^{d\over
2}\over(x/2)^{\frac{d-2}{2}}}\int_{0}^{\infty}{dpp^{\frac{d}{2}}\over(2\pi)^{d}}J_{\frac{d-2}{2}}(px)\widetilde{p}^{\nu}K_{\nu}\left(2\widetilde{p}\right)$
(4) $\displaystyle=$
$\displaystyle\frac{\Gamma(\nu+\frac{d}{2})}{\Gamma(\nu)}\left(\frac{\alpha}{4\pi\eta}\right)^{\frac{d}{2}}\left(1+{\alpha
x^{2}\over 4\eta}\right)^{-(\nu+\frac{d}{2})},$

where $J_{\mu}(x)$ is the Bessel function of the first kind and in the second
equation we used a formula found in Gradshteyn:2014aa

$\displaystyle\int_{0}^{\infty}\\!\\!\\!dpp^{\bar{\mu}+\bar{\nu}+1}J_{\bar{\mu}}(ap)K_{\bar{\nu}}\left(bp\right)={(2a)^{\bar{\mu}}(2b)^{\bar{\nu}}\over(a^{2}+b^{2})^{\bar{\mu}+\bar{\nu}+1}}\Gamma(\bar{\mu}+\bar{\nu}+1)$

for
$\mathrm{Re}(b)>|\mathrm{Im}(a)|,\mathrm{Re}(\bar{\mu})>|\mathrm{Re}(\bar{\nu})|-1$.
This implies $\nu<d/2$. On the other hand, $\rho_{\eta}(x)\to\delta^{d}(x)$ in
the limit $\eta\to 0$ as long as $\nu>0$. Thus the new parameters are
restricted so that

$0<\nu<d/2.$ (5)

One can check that the limit of $\alpha\to 0$ reduces to the case of the free
flow. In the limit, $\nu$ goes to infinity, and with the help of the Stirling
formula we find

$\displaystyle\lim_{\alpha\to
0}\rho_{\eta}(x)|_{\beta=1}=\left(\frac{1}{4\pi\eta}\right)^{\frac{d}{2}}e^{-x^{2}\over
4\eta},$ (6)

which is indeed the smearing function of the free flow (1).

###  II.2 2-2. Bulk information metric

The flow equation method provides a way to compute a bulk (or
$d+1$-dimensional) metric in the expected holographic space without assuming
holographic geometry Aoki:2017bru ; Aoki:2015dla . This can be done by
constructing an operator in CFT called a metric operator such that

$\displaystyle\hat{g}_{MN}(x;\eta)$ $\displaystyle:=$
$\displaystyle\ell^{2}\sum_{a=1}^{N}{\partial\sigma^{a}(x;\eta)\over\partial
X^{M}}{\partial\sigma^{a}(x;\eta)\over\partial X^{N}}.$ (7)

Here $X^{M}=(x^{\mu},z)$, where $z=\sqrt{\eta/\gamma}$ with $\gamma$ a
constant, $\sigma^{a}$ is a normalized smeared field defined by
$\sigma^{a}(x;\eta)=\frac{1}{\sqrt{\left\langle\phi(x;\eta)^{2}\right\rangle}}\phi^{a}(x;\eta)$
with $\phi(x;\eta)^{2}=\sum_{b=1}^{N}\phi^{b}(x;\eta)^{2}$, and $\ell$ is a
length scale chosen by hand. Then the VEV of the metric operator yields a
metric in the expected holographic space:
$g_{MN}(X)=\left\langle\hat{g}_{MN}(x;\eta)\right\rangle$.

Comments on this construction are in order. Firstly this is a well-defined
operator since the smearing by the flow equation removes the contact
divergence. Indeed the two-point function of the smeared field is evaluated at
the coincident point as
$\left\langle\sum_{a}\phi^{a}(x;\eta)^{2}\right\rangle=NC_{1}z^{-2\Delta}$,
where $C_{1}=\gamma^{-\Delta}\int
d^{d}y_{1}d^{d}y_{2}\rho_{1}\left(y_{1}\right)\rho_{1}\left(y_{2}\right)\frac{C_{0}}{|y_{1}-y_{2}|^{2\Delta}}.$

Secondly due to this construction the classical geometry emerges after taking
quantum average as well as the large $N$ limit Aoki:2015dla . This can be seen
from the behavior of the VEV of the Einstein tensor operator
$G_{MN}(\hat{g}_{PQ})$ Aoki:2018dmc

$\displaystyle\langle G_{MN}(\hat{g}_{PQ})\rangle$ $\displaystyle=$
$\displaystyle G_{MN}(\langle\hat{g}_{PQ}\rangle)+O\left({1/N}\right)$ (8)

due to the large $N$ factorization. The leading term gives the classical AdS
value, while the correction term is interpreted as the quantum correction in
the bulk.

Thirdly the induced metric $g_{MN}(X)$ can be interpreted as the Bures
information metric for a mixed state defined by
$\rho(X)=\sum_{a=1}^{N}|\sigma^{a}(x;\eta)\rangle\langle\sigma^{a}(x;\eta)|$,
where $|\sigma^{a}(x;\eta)\rangle=\sigma^{a}(x;\eta)|0\rangle$ Aoki:2017bru .
Note that using the normalized field $\sigma^{a}(x;\eta)$ is important to
ensure $\mathrm{tr}\rho=1$.

It was confirmed in the previous study that the induced metric takes the
standard form of the AdS one in the Poincaré patch for the free flow (1), and
it becomes the standard one also for the interpolating flow (2) by a proper
choice of $\gamma$, which corresponds to a redefinition of the radial
coordinate $z$. Thus, as shown in Aoki:2019dim , a modification of a flow
equation can be interpreted as a different choice of the coordinate system (or
gauge) in the bulk. As in the case of the free flow, however, a general
interpolating flow is not helpful to figure out the bulk interpretation of a
CFT excited state. In the next subsection, employing a symmetry argument, we
search for a special value of $\alpha$ and $\beta$ by which the bulk
interpretation becomes transparent.

###  II.3 2-3. Special flow

An infinitesimal conformal transformation for a primary scalar field
$\varphi^{a}(x)$ is

$\displaystyle\delta^{\rm conf}\varphi^{a}(x)=$ $\displaystyle-{\delta
x}^{\mu}\partial_{\mu}\varphi^{a}(x)-{\Delta\over d}(\partial_{\nu}\delta
x^{\nu})\varphi^{a}(x),$ (9)

where $\delta x^{\mu}=a^{\mu}+\omega^{\mu}{}_{\nu}x^{\nu}+\lambda
x^{\mu}+b^{\mu}x^{2}-2x^{\mu}(b\cdot x)$. Then the conformal transformation
for the normalized smeared field is given by

$\displaystyle\delta^{\rm conf}\sigma^{a}(x;\eta)=$
$\displaystyle\frac{1}{\sqrt{\left\langle\phi(x;\eta)^{2}\right\rangle}}\int
d^{d}y\rho_{\eta}(x-y)\delta^{\rm conf}\varphi^{a}(y)$ $\displaystyle=$
$\displaystyle\delta^{\rm diff}\sigma^{a}(x;\eta)+\delta^{{\rm
extra}}\sigma^{a}(x;\eta),$ (10)

where we decomposed it into the $d+1$-dimensional AdS isometry $\delta^{\rm
diff}\sigma^{a}$ written as

$\displaystyle\delta^{\rm diff}\sigma^{a}(x;\eta)=$
$\displaystyle-{\bar{\delta}x}^{\mu}\partial_{\mu}\sigma^{a}(x;\eta)-\bar{\delta}z\partial_{z}\sigma^{a}(x;\eta)$
(11)

with $\bar{\delta}x^{\mu}=\delta x^{\mu}+\frac{4\gamma}{\alpha}z^{2}b^{\mu}$,
$\bar{\delta}z=(\lambda-2b\cdot x)z$, and the rest

$\displaystyle\delta^{{\rm
extra}}\sigma^{a}(x;\eta)={2(\nu-\frac{d}{2}+\Delta)\over\sqrt{\left\langle\phi(x;\eta)^{2}\right\rangle}}\int
d^{d}y\,b\cdot y\rho_{\eta}(y)\varphi^{a}(x-y).$

Therefore, $\delta^{{\rm extra}}\sigma^{a}(x;\eta)$ vanishes for all conformal
transformations if and only if $\nu=\frac{d}{2}-\Delta$. In this case the
conformal transformation directly converts into the general coordinate
transformation on $\sigma^{a}$ in the bulk with an extra coordinate $z$, which
becomes the AdS isometry for $\gamma=\alpha/4$. Thus the conformal invariance
guarantees that the bulk geometry is AdS as $g_{MN}(X)={R_{\rm AdS}^{2}\over
z^{2}}\delta_{MN}$ with $R_{\rm AdS}=\ell\sqrt{\frac{\Delta(d-\Delta)}{d+1}}$.
We refer to this flow with the special parametrization as the special flow.

For the special flow, the smearing function for $\sigma^{a}$ is given by
$\sigma^{a}(x;\eta)=\int d^{d}y\,K(X,y)\varphi^{a}(y)$, where

$\displaystyle
K(X,y)={\rho_{\eta}(x-y)\over\sqrt{\left\langle\phi(x;\eta)^{2}\right\rangle}}=C_{2}\left({z\over
z^{2}+(x-y)^{2}}\right)^{d-\Delta}$ (12)

with
$C_{2}=\frac{\Gamma(d-\Delta)}{\Gamma(d/2-\Delta)}\sqrt{(\alpha/4)^{\Delta}\over
N\pi^{d}C_{1}}$. This kernel is in fact formally the same as the boundary to
bulk propagator Witten:1998qj . Thus this result in some sense reproduces the
standard well-known result of the AdS/CFT correspondence only by symmetry
argument, though there is a conceptually important difference between the
standard bulk reconstruction and the one by means of flow equations. That is,
in the former approach the smeared objects are singlet or gauge invariant,
while in the latter they are elementary fields of the theory. Accordingly, the
dimension of a smeared object in the latter approach is less than ${d\over 2}$
following from (5), which is complementary to the condition in the standard
bulk reconstruction that the dimension is greater than $d-1$. (See
Hamilton:2006az  for instance.)

Then a natural question is how singlet operators in CFT are realized in the
bulk to satisfy the equation of motion. This question is addressed in the next
section.

##  III 3\. Bulk reconstruction for excited states

In the previous section we confirmed not only that the bulk geometry
corresponding to a CFT vacuum emerges as AdS but also that the conformal
symmetry properly acts as the AdS isometry on the flowed scalar field
$\sigma^{a}$ for the special flow. In this section we extend this bulk
reconstruction to an excited state by a singlet primary scalar operator in
CFT.

###  III.1 3-1. Symmetry constraint

A key to this extension is that the specially flowed operator $\sigma(x;\eta)$
is constructed so that the conformal transformation acts on it just as the AdS
isometry with $z=\sqrt{4\eta/\alpha}$ as the AdS radial coordinate. Therefore
a correlation function in terms of the specially flowed field and CFT primary
operators is constrained severely by conformal symmetry. To see this
concretely, we consider the following correlation function

$\displaystyle\left\langle G_{M_{1}M_{2}\cdots
M_{n}}[\sigma](X)T_{\nu_{1}\nu_{2}\cdots\nu_{p}}(y)\right\rangle,$ (13)

where $G_{M_{1}M_{2}\cdots M_{n}}[\sigma]$ is a CFT operator constructed with
the specially flowed operator $\sigma$ while $T_{\nu_{1}\nu_{2}\cdots\nu_{p}}$
is a primary tensor operator in CFT. Note that these operators are not
restricted to singlet ones. Then the conformal symmetry constrains this
correlation function such that

$\displaystyle\left\langle G_{M_{1}M_{2}\cdots
M_{n}}[\sigma](\tilde{X})T_{\mu_{1}\mu_{2}\cdots\mu_{p}}(\tilde{y})\right\rangle$
$\displaystyle=$ $\displaystyle{\partial
X^{N_{1}}\over\partial\tilde{X}^{M_{1}}}\cdots{\partial
X^{N_{n}}\over\partial\tilde{X}^{M_{n}}}\times J(y)^{-\Delta_{T}}{\partial
y^{\nu_{1}}\over\partial\tilde{y}^{\mu_{1}}}\cdots{\partial
y^{\nu_{p}}\over\partial\tilde{y}^{\mu_{p}}}$ $\displaystyle\times\left\langle
G_{N_{1}N_{2}\cdots
N_{n}}[\sigma](X)T_{\nu_{1}\nu_{2}\cdots\nu_{p}}(y)\right\rangle,$ (14)

where $\widetilde{y},\widetilde{X}$ are a finite conformal transformation for
$y$, an AdS isometric one for $X$, respectively,
$J(y)=|\mathrm{det}(\partial_{\nu}\widetilde{y}^{\mu})|^{1/d}$, $\Delta_{T}$
is the conformal dimension of $T$. A generalization to $k$ bulk operators and
$s$ boundary operators is straightforward. This class of correlation functions
is systematically studied in the context of the AdS/CFT correspondence
Papadimitriou:2004ap ; Costa:2011dw ; Costa:2011mg ; Aharony:2020omh .

As a first application, let us compute the correlation function of the smeared
operator and the unsmeared one. In order to satisfy the constraint (14), the
position dependence is completely fixed as

$\left\langle\sigma^{a}(x;\eta)\varphi^{b}(y)\right\rangle=c\delta^{ab}\left(\frac{z}{(x-y)^{2}+z^{2}}\right)^{\Delta},$
(15)

where the constant $c$ is fixed by rewriting this in terms of the unnormalized
smeared operator as

$\left\langle\phi^{a}(x;\eta)\varphi^{b}(y)\right\rangle=\delta^{ab}\frac{c\sqrt{NC_{1}}}{((x-y)^{2}+z^{2})^{\Delta}}.$
(16)

Taking the limit $\eta\to 0$, the left-hand side reduces to the given two-
point correlator of the elementary field if and only if
$c=C_{0}/\sqrt{NC_{1}}$. Thus eqs.(12) and (15) give

$\displaystyle\left\langle\sigma^{a}(x_{1};\eta_{1})\sigma^{b}(x_{2};\eta_{2})\right\rangle\propto$
$\displaystyle\delta^{ab}\int
d^{d}y\left[\frac{z_{1}}{(x_{1}-y)^{2}+z_{1}^{2}}\right]^{\Delta}\left[\frac{z_{2}}{(x_{2}-y)^{2}+z_{2}^{2}}\right]^{d-\Delta}.$

Note that this can be written by using a shadow operator of $\varphi^{c}$
denoted by $(\varphi^{c})^{\star}$, whose conformal dimension is
$\Delta^{\star}=d-\Delta$ as

$\displaystyle\left\langle\sigma^{a}(x_{1};\eta_{1})\sigma^{b}(x_{2};\eta_{2})\right\rangle\propto$
$\displaystyle\sum_{c=1}^{N}\int
d^{d}y\left\langle\sigma^{a}(x_{1};\eta_{1})|\varphi^{c}(y)\right\rangle\left\langle\varphi^{c}(y)^{\star}|\sigma^{b}(x_{2};\eta_{2})\right\rangle.$

Taking into account the normalization this can be computed as

$\displaystyle\left\langle\sigma^{a}(x_{1};\eta_{1})\sigma^{b}(x_{2};\eta_{2})\right\rangle=\frac{\delta^{ab}}{N}F_{12}(X_{1},X_{2}),$
(17) $\displaystyle
F_{12}(X_{1},X_{2}):={}_{2}F_{1}\left(\frac{\Delta}{2},\frac{d-\Delta}{2};\frac{d+1}{2};1-\frac{{\cal
R}^{2}}{4}\right),$

where ${\cal R}=\frac{(x_{1}-x_{2})^{2}+z_{1}^{2}+z_{2}^{2}}{z_{1}z_{2}}$ is
the SO($1,d+1$) invariant ratio. This explicitly shows the absence of the
contact singularity of the smeared operator.

In the next subsection we shall see that this technique is also useful for the
bulk reconstruction for excited states.

###  III.2 3-2. GKP-Witten relation

In the framework using flow equations, excitations in the bulk are
reconstructed from the corresponding excited state via pre-geometric operators
Aoki:2018dmc . To see this concretely, let us consider a state excited by a
primary scalar singlet operator denoted by $S(y)$ and its dimension by
$\Delta_{S}$. (In O$(N)$ vector models,
$S(y)\propto\sum_{a}\text{:}\varphi^{a}\varphi^{a}\text{:}(y)$.) Corresponding
to the boundary excitation, the bulk scalar field is expected to be excited.
If we denote it by $\chi(X)$, the excitation is described by
$\chi(X)=\sum_{a}\langle 0|\sigma^{a}(x;\eta)^{2}S(y)|0\rangle.$ Let us prove
that the field $\chi(X)$ satisfies the equation of motion on the emergent AdS
background. This can be seen by determining the position dependence similar to
(15) implied by the symmetry constraint (14):

$\chi(X)=C_{S}\left({z\over(x-y)^{2}+z^{2}}\right)^{\Delta_{S}},$ (18)

where the overall constant $C_{S}$ remains undetermined by the symmetry alone.
This can be explicitly seen for the case of the vector model by taking the
large $N$ limit, which leads to $\Delta_{S}=2\Delta_{\varphi}$ due to the
large $N$ factorization. From this expression it is easy to see that $\chi(X)$
satisfies the equation of motion with the expected mass dependence $(\Box_{\rm
AdS}-m_{S}^{2})\chi(X)=0$, where ${\Box_{\rm
AdS}}:=(z^{2}(\partial_{z}^{2}+\partial^{2})-(d-1)z\partial_{z})$ is the
d’Alembertian on the AdS space, ${m_{S}^{2}}=\Delta_{S}(\Delta_{S}-d)$.

In addition it is not difficult to reproduce the the GKP-Witten relation
Gubser:1998bc ; Witten:1998qj  by coupling the field $\chi(X)$ to a
sufficiently small source field $J(y)$ and integrating all over the space:

$\chi_{J}(X)=\sum_{a=1}^{N}\langle 0|\sigma^{a}(x;\eta)^{2}\int
d^{d}yJ(y)S(y)|0\rangle.$ (19)

Then by using (18) the asymptotic behavior of the field $\chi_{J}(X)$ at the
boundary is given by $\chi_{J}(X)\to z^{\Delta_{S}}\left\langle
S(x)\right\rangle_{J},$ up to an overall constant, where $\left\langle
S(x)\right\rangle_{J}=\left\langle S(x)e^{\int
d^{d}yJ(y)S(y)}\right\rangle\sim\int d^{d}yJ(y)\left\langle
S(x)S(y)\right\rangle$.

##  IV 4\. Discussion

We emphasize that there are conceptual differences between the flow method and
the standard bulk reconstruction. (See Terashima:2017gmc  for a different
point of view on the standard bulk reconstruction.) Although both methods
employ formally the same smearing function to construct the bulk field from
the boundary field, they are in fact complementary to each other. (i) The
smearing in the flow method is applied to the elementary non-singlet field
$\varphi^{a}$ in the Euclidean path integral, while the bulk reconstruction
gives a relation between boundary and bulk free local singlet quantum
operators with Lorentzian signature. (ii) In the bulk reconstruction, the
boundary CFT primary field must have a conformal dimension larger than $d-1$.
(This constraint can be loosened but then the formula needs to be
modifiedAoki:2021ekk ; bhowmick2017bulk .) On the other hand, $\varphi^{a}$ in
the flow equation must have $\Delta\leq d/2$. (iii) The VEV of the metric
field realizes an emergent bulk AdS geometry from the CFT in the flow method.
On the other hand, the existence of a background AdS spacetime is assumed from
the beginning in the bulk reconstruction.

The smeared and normalized field $\sigma^{a}$ generates not only the bulk
geometry but also other degrees of freedom as composite singlet fields in the
bulk, similarly to QCD where all hadrons are made of quarks. In the bulk
construction by the flow method, the constraint (14) imposed by the symmetry
is more fundamental than the metric, which is secondary and is fixed by the
constraint. The symmetry and its constraint hold at the fully quantized level
and at all orders in the large $N$ expansion.

It would be interesting to ask how scalar excited state contributions affect
the AdS geometry described by the VEV of the metric field. As before, we
consider the VEV of the metric operator in the presence of a small $J(0)$ as

$\displaystyle\bar{g}_{MN}^{J}(X)\simeq\langle
0|\hat{g}_{MN}(x;\eta)|0\rangle+J(0)\langle 0|\hat{g}_{MN}(x;\eta)|S\rangle.$
(20)

The constraint (14) leads to

$\displaystyle\langle 0|\hat{g}_{MN}(X)|S\rangle$ $\displaystyle=$
$\displaystyle{T^{\Delta_{{S}}}\over
z^{2}}\left[a_{1}\delta_{MN}+a_{2}T_{M}T_{N}\right],$ (21)

where $T:={z\over x^{2}+z^{2}},\ T_{z}:={x^{2}-z^{2}\over x^{2}+z^{2}},\
T_{\mu}:=-{2x_{\mu}z\over x^{2}+z^{2}},$ and the undetermined constants are
fixed for the free $O(N)$ vector model as $a_{1}=0,\ a_{2}={\sqrt{{2\over
N}}\frac{\Delta^{2}\ell^{2}\Gamma(d)}{\Gamma({d\over 2}+1)\Gamma({d\over
2})}},\ \Delta:={d-2\over 2}.$ Thus the metric $\bar{g}_{MN}^{J}(X)$ describes
an asymptotically AdS space.

Since the above metric $\bar{g}_{MN}^{J}(X)$ deviates from the pure AdS
metric, there must exist a matter energy momentum tensor in the bulk, which
can be evaluated through the Einstein equation as

$\displaystyle 2\kappa T_{MN}^{(S)}$ $\displaystyle=$
$\displaystyle{Ja_{2}(d-1)\over R_{\rm AdS}^{2}}{T^{2\Delta}\over
z^{2}}\left[\left(\Delta-{d\over 2}\right)\delta_{MN}-\Delta
T_{M}T_{N}\right].$

We need to find a bulk matter theory which realizes this metric and the
corresponding energy momentum tensor, analogously to what we have found for a
non-relativistic CFTAoki:2019dim , but leave this issue for future studies.

The constraint (14) reproduces bulk to boundary 2-pt functions in the AdS/CFT
correspondence not only for a scalar field but also for an arbitrary bulk
field such as a tensor. However, there exists an issue on the bulk to bulk
2-pt functions constructed with this method. For example, let us compute the
2-pt function of the singlet scalar $S(x;z)$: $\langle
S(x_{1};z_{1})S(x_{2};z_{2})\rangle\\\ =1+{2\over
N}F^{2}_{12}(X_{1},X_{2})+\langle S(x_{1};z_{1})S(x_{2};z_{2})\rangle_{c},$
where the connected part is evaluated as

$\displaystyle\langle
S(x_{1};z_{1})S(x_{2};z_{2})\rangle_{c}=\left(\prod_{i=1}^{4}\int
d^{d}y_{i}\right)K(X_{1},y_{1})K(X_{1},y_{2})$ $\displaystyle
K(X_{2},y_{3})K(X_{2},y_{4})\langle\varphi^{a}(y_{1})\varphi^{a}(y_{2})\varphi^{b}(y_{3})\varphi^{b}(y_{4})\rangle_{c}.$

If the boundary theory is free, the bulk to bulk 2-pt function has no
singularity at $X_{1}=X_{2}$. This shows that the bulk theory is not described
by point-like objects if the boundary CFT is free or weakly interacting. On
the other hand, a singularity is expected to emerge in the strong coupling
limit of CFT, since the AdS/CFT correspondence predicts that a bulk theory
becomes local in this limit. It would be interesting to evaluate the connected
part explicitly in an interacting theory such as the $O(N)$ vector model.

## Acknowledgement

This work is supported in part by the Grant-in-Aid of the Japanese Ministry of
Education, Sciences and Technology, Sports and Culture (MEXT) for Scientific
Research (Nos. JP16H03978, JP18K03620, JP18H05236, JP19K03847, JP22H00129),
and by the NKFIH grant K134946. J.B. acknowledges support from the
International Research Unit of Quantum Information (QIU) of Kyoto University
Research Coordination Alliance, and also would like to thank the Yukawa
Institute for Theoretical Physics at Kyoto University, where most of this work
has been carried out, for support and hospitality by the long term visitor
program. T. O. would like to thank YITP for their kind hospitality during his
stay for the sabbatical leave from his home institute.

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