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We explore the construction of the dual bulk theory in the flow equation approach. We compute the vacuum expectation value of the Einstein operator at the next-to-leading order in the

Two decades have passed since the anti-de Sitter / conformal field theory (AdS/CFT) correspondence was discovered [

One of the recent focuses in the study of the AdS/CFT correspondence is on how diffeomorphism invariance is encoded in a boundary CFT and the Einstein equation is reproduced from boundary data. Such a study was initiated at the linear order of perturbation around the AdS background by using the entanglement entropy [

In this setting, this paper aims to propose a new scheme to compute bulk dynamical observables from a boundary CFT by employing a new approach to AdS/CFT incorporating a flow equation [

The rest of the paper is outlined as follows. In Sect.

In this section we propose a method to compute dual observables in the flow equation approach. In this approach we generally construct

In this subsection we illustrate how to construct a metric operator in the flow equation approach; see also Ref. [

Here,

To approach the dual AdS geometry from CFT, it is convenient to consider the free flow, which is realized by choosing

The solution is given by

The free flow equation is formally the same form as the heat equation, so that the smeared operator is given by superposing all the original operators inserted at each point of the space with the Gaussian weight whose standard deviation is the smearing scale

Following the standard renormalization group transformation procedure, we (re)normalize the smeared field

Note that this operator is well defined due to the fact that the flowed operators are free from the contact singularity.

We can introduce the metric operator, which becomes the metric in the holographic space after taking the quantum average, as

It was shown in Ref. [

Once the metric operator is constructed, other pre-geometric operators are defined by replacing the metric that appears in the definition of the corresponding (classical) geometric object with the metric operator. For example, the Levi–Civita connection operator is defined by

In this section we evaluate the VEV of the Einstein tensor operator for a free

Since the Einstein tensor is now evaluated on the vacuum, it is natural to think that the corresponding bulk stress–energy tensor consists only of the cosmological constant term:

In what follows, we compute the cosmological constant

Let us first compute induced geometric observables for a free

In the current case,

The VEV of the induced metric becomes

Using the factorization in the large-

Induced curvatures at the LO are computed as

Therefore, the cosmological constant of the dual geometry is evaluated through Eq. (

We proceed to the next-to-leading order computation of the induced Einstein tensor. For this purpose we employ a covariant perturbation expansion: that is, we expand an arbitrary operator

Terms with increasing numbers of

We summarize the results of this expansion in

For a free ^{1}

This final result is manifestly covariant, even though the calculation in the intermediate step contains non-covariant terms. This is a non-trivial check of our result.

As a result, the induced Einstein tensor evaluated at the vacuum is given by

As asserted at the beginning of this section, this quantity is related to the bulk stress energy tensor through the bulk Einstein equation, Eq. (

Since

Finally, we discuss our results in terms of the conformal symmetry or AdS isometry. In the previous publications [

For this purpose, as in Refs. [

To this end, explicit expressions for

Here,

We need to consider

Equation (

These guarantee that each term is covariant under isometry, and thus proportional to

In this paper, we have constructed the holographic space from the primary scalar field in a free massless

In our approach, the bulk AdS radial direction emerges as the smearing scale for the boundary CFT. It is important to clarify how to determine the whole structure of the bulk theory. The bulk stress–energy tensor corresponding to the vacuum state calculated in this paper may give a hint to constructing the bulk theory.

In this paper we computed the one-loop correction to the cosmological constant in the bulk theory, which is supposed to be the free higher-spin theory [

The next important step is to evaluate the bulk stress–energy tensor corresponding to excited states. Indeed, we can easily generalize the computation of the VEV for the Einstein operator presented in this paper to that of arbitrary states as follows. We consider a set of states

Notice that we have already calculated the first term in Eq. (

It is very important to compute this bulk stress–energy tensor in the construction of the dual bulk theory beyond the vacuum or geometry level. We are currently calculating

This program can be extended to the case of the

We hope to report on the progress with these issues in the near future.

S. A. is supported in part by a Grant-in-Aid of the Japanese Ministry of Education, Sciences and Technology, Sports and Culture (MEXT) for Scientific Research (no. JP16H03978), by a priority issue (Elucidation of the fundamental laws and evolution of the universe) to be tackled by using Post “K” Computer, and by the Joint Institute for Computational Fundamental Science (JICFuS). This work was partially supported by the Hungarian National Science Fund Országos Tudományos Kutatási Alapprogramok (OTKA) (under K116505).

Open Access funding: SCOAP

We introduce the fluctuation of the metric operator around its VEV as

We then expand the Levi–Civita connection as

Similarly, we have

Riemann curvatures are expanded as

Using Eqs. (

At the next-to-leading order, the Einstein tensor is evaluated as

In this appendix we calculate various two-point functions of

We define

Derivatives of

The simplest one can be calculated from

Similarly,

We now evaluate derivatives of

Combining the above results, we finally obtain

^{1}This result was checked in a slightly different computation by