Flow equation for the large $N$ scalar model and induced geometries

We study the proposal that a $d+1$ dimensional induced metric is constructed from a $d$ dimensional field theory using gradient flow. Applying the idea to the O($N$) $\varphi^4$ model and normalizing the flow field, we have shown in the large $N$ limit that the induced metric is finite and universal in the sense that it does not depend on the details of the flow equation and the original field theory except for the renormalized mass, which is the only relevant quantity in this limit. We have found that the induced metric describes Euclidean Anti-de-Sitter (AdS) space in both ultra-violet (UV) and infra-red (IR) limits of the flow direction, where the radius of the AdS is bigger in the IR than in the UV.


Introduction
The AdS/CFT (or more generally Gravity/Gauge theory) correspondence [1] is a surprising but significant finding in field theories and string theories. The original correspondence claims that a d dimensional conformal field theory is equivalent to some d + 1 dimensional super-gravity theory on an AdS background. After the first proposal, there appeared many pieces of evidence which indicate that the correspondence is true, and applications to various areas in physics, such as AdS/QCD or AdS/CMP, have been successfully investigated.
Even though the AdS/CFT correspondence might be explained by the closed string/open string duality, the claimed equivalence is still mysterious and a deeper understanding on the correspondence is necessary.
In a previous paper [2], two of the present authors and their collaborator tried to understand the gravity/field theory correspondences from a different point of view, proposing an alternative method to define a geometry from a field theory: A quantum field theory in d dimensions is lifted to a d + 1 dimensional one using gradient flow [3][4][5][6], where the flow time t, which represents the energy scale of the original d dimensional theory, is interpreted as an additional coordinate. Then the induced metric is defined from this d + 1 dimensional field.
As the metric is derived from the original d dimensional theory together with its scale dependence, the method proposed in the paper can be applied in principle to all field theories. As a concrete example, the method was applied to the O(N ) non-linear sigma model (NLSM) in two dimensions, and the vacuum expectation value (VEV) of the three dimensional induced metric was shown to describe an AdS space in the massless limit. 1 There are two key aspects in this proposal. First of all, the d + 1 dimensional induced metric becomes classical in the large N limit, and the quantum corrections can be calculated order by order in the large N expansion. Secondly, in the example mentioned above, the three dimensional induced metric, constructed from a product of three dimensional flow fields at the same point, is free from UV divergences in the large N limit. Instead, if the original two dimensional fields were used directly to define the two dimensional metric, it would badly diverge. These two special properties allow us to infer that (the VEV of) the induced metric describes a geometry.
In this paper, we further investigate this proposal, explicitly considering a more general model, the O(N ) invariant ϕ 4 model in d dimensions, which can describe the free scalar model as well as the NLSM as limiting cases. Furthermore, as a generalization of the proposal in Ref. [2], we consider the case that parameters (mass and coupling) are different in the original d dimensional theory and the flow equation, in order to see how the behavior of the flow field depends on the choice. We also introduce a normalization of the flow field, to interpret the gradient flow as a renormalization group (RG) transformation. Our normalization condition for the flow field requires a rescaling factor which becomes unity in the NLSM limit, so that the result in the previous paper [2] is unchanged. We then define the induced metric from this normalized flow field. It turns out that the metric defined in this way leads to the same geometry, irrespective of parameters in the flow equation as well as in the original d dimensional theory. We show that the d + 1 dimensional space described by the metric becomes AdS both in UV and IR limits as long as d > 2.
This paper is organized as follows. In Sec. 2, we first briefly summarize the proposal in Ref. [2], together with a few modifications introduced in this paper. We explain some properties of the d dimensional model in Sec. 3, and solve the flow equation in the large N limit in Sec. 4. We calculate the induced metric in Sec. 5 and the Einstein tensor in Sec. 6, showing that the space defined by the metric becomes AdS in both UV and IR limits. The behavior of the Einstein tensor between two limits is also investigated in Sec. 6. We finally give a summary of this paper in Sec. 7. In appendix A, we show that a divergence of the flow field appears for the ϕ 4 theory in perturbation theory but it disappears non-perturbatively in the large N limit. Some properties of the incomplete gamma function used in the main text are given in appendix B.

Proposal
In this section, we briefly summarize the proposal of Ref. [2] and explain some modifications introduced in this paper.
In this paper we consider the large N scalar field ϕ a (x) with the d dimensional space-time coordinate x and the large N index a = 1, 2, · · · , N , whose action S describes the ϕ 4 model and will be explicitly given in the next section.
The d dimensional field ϕ a (x) is extended to φ a (t, x) in d + 1 dimensions, using the gradient flow equation as [3,4] with the initial condition φ a (0, x) = ϕ a (x), where the flow action S in this paper can be different from the action S in the original d dimensional theory. Since the length dimension of t is 2 and t ≥ 0, a new variable τ = 2 √ t is introduced. We then denote the d + 1 dimensional coordinate as z = (τ, x) ∈ R + (= [0, ∞]) × R d and the flow field as φ a (z).
In Ref. [2], a d + 1 dimensional metric was defined aŝ where h > 0 is a dimensionful constant. In this paper, however, we modify this definition slightly toĝ where σ a is the normalized flow field defined by which is equivalent to a constant renormalization, as we will see later, and h is a constant with the mass dimension −2. The average φ 2 (z) in the above definition will be defined in the next paragraph. The reason why we use σ a instead of φ a to define the metric is as follows. The RG transformation consists of two procedures: first UV modes are integrated out and then the field is normalized for the RG flow to have a fixed point. Since the gradient flow itself corresponds to the former procedure only, a field normalization is introduced as in eq. (4). One may adopt a different field normalization, but eq. (4) leads to interesting results as will be seen in later sections. We call this condition the NLSM normalization, as The aboveĝ µν (z) is an induced metric on a d + 1 dimensional manifold R + × R d from some manifold in R N defined by σ a (z), which becomes an N − 1 dimensional sphere after the quantum average as σ 2 (z) = 1. Here the expectation values ofĝ µν (and its correlations) are defined as where O S is the expectation value of O(ϕ) in d dimensions with the action S as Even though the composite operatorĝ µν (z) contains a product of two local operators at the same point z, ĝ µν (z) is finite as long as τ = 0 for the two dimensional NLSM in the large N limit [2]. Although the finiteness of the flow field was proven for gauge theories [9] and some scalar theories [10] (see also the latest extension in [11]), it is not guaranteed in general. Indeed it was pointed out recently that the flow field of the ϕ 4 model gives an extra UV divergence in perturbation theory [12]. (The modified flow equation can avoid this divergence. [13]) As shown later, however, the induced metricĝ µν in eq. (3) is free from such UV divergences in the large N limit. In appendix A, we explicitly demonstrate that a divergence indeed appears in perturbation theory, but it disappears non-perturbatively in the large N limit. On the other hand, if the d dimensional induced metric were defined from the d dimensional field ϕ, it would diverge badly, and hence a geometry could not be defined from it.
Moreover, thanks to the large N factorization, quantum fluctuations of the metricĝ µν are suppressed in the large N limit. The n-point correlation function ofĝ µν behaves as showing that the induced metricĝ µν becomes classical in the large N limit, and quantum fluctuations are sub-leading and calculable in the large N expansion. This property allows a geometrical interpretation of the metricĝ µν . In Sec. 6, the VEV of the Einstein tensor is calculated directly from ĝ µν , as in the classical theory.
3 Large N model

Model
In this paper, we consider the N component scalar ϕ 4 model in d dimensions, defined by the action where ϕ a (x) is an N component scalar field, ( · ) indicates an inner product of N component , µ 2 is the bare scalar mass parameter, and u is the bare coupling constant of the ϕ 4 interaction, whose canonical dimension This model describes the free massive scalar at u = 0, while it is equivalent to the NLSM in the u → ∞ limit, whose action is obtained from eq. (9) as with the replacement In addition, the flow time has to be also rescaled by λ for the flow equation in this limit.

Calculation in the large N limit
The partition function is evaluated as [14] Z where the effective action is given by with for arbitrary functions F (x) and G(x).
The large N limit corresponds to the saddle point, 2iβ(x) = m 2 , determined by the saddle point equation as where Z(m) = dp 1 which is divergent at d > 1. We therefore introduce some regularization such as lattice, dimensional or momentum cut-off regularization, but do not specify it unless necessary. In this paper, sending the cut-off to infinity is called "the continuum limit".
The two point function is then evaluated as at u = ∞, where m is regarded as the renormalized mass. Therefore µ 2 must be tuned to keep m 2 finite in the continuum limit. The saddle point equation, says that µ 2 → −∞ in the continuum limit at d > 1 as long as u > 0.
In the NLSM limit u → ∞, we have Therefore λ → 0 in the continuum limit at d > 1.
4 Flow equation and its solution in the large N limit

Flow equation
In this paper, we consider the flow equation, given by where µ 2 f and u f can be different from µ 2 and u in the original d-dimensional theory. The flow with µ f = µ and u f = u is called gradient flow, as it is given by the gradient of the original action.

The solution in the large N limit
In the large N limit, the solution has the following form in momentum space [15], The flow equation (21) in the large N limit leads to the equation for f (t): where Using X(t) = f −2 (t), the equation becomeṡ so that the solution is given by Introducing m f by the relation the solution is further reduced as where In the case of the interacting flow with for t > 0.
For the flow with u f = 0, which we call "the free flow", we simply have f (t) = e −m 2 f t for all t ≥ 0.

Two-point function
The two-point function of the flow field with u f > 0 is given by In the continuum or NLSM limit, we have which only depends on renormalized quantities m f and m, but does not depend on bare parameters, µ 2 f , u f , µ 2 and u. Note however that Z(m f ) diverges in the continuum limit for d > 1.
If we take the NLSM limit for the flow equation (u f → ∞) but without taking the continuum limit, the two-point function of σ a (t, This two-point function is finite even in the continuum limit, as shown explicitly at d = 2 [15,16]. At u f = 0, on the other hand, we obtain which is of course manifestly finite.

Normalized flow field
As mentioned before, to construct a RG transformation using the flow equation, which is merely a kind of smearing procedure for UV fluctuations, we introduce a normalization condition for the flow field.
There is no unique way to define the normalized flow field. In the standard block spin transformation, for example, the normalization factor for the field is so chosen so that some fixed point can appear for the defined RG transformation. In this paper, we propose the following normalization for the flow field: so that the normalized flow field is dimensionless and satisfies the non-linear constraint σ 2 (t, x) = 1. Because of this property, we may call it the NLSM normalization.
For the interacting flow (u f = 0), since we have which gives σ a (t, x)σ b (s, y) = δ ab N 1 ζ 0 (t)ζ 0 (s) dp e −(t+s)p 2 e ip(x−y) (µ 2 f , u f , µ 2 , u) as well as the renormalized flow mass m f even without taking the continuum or NLSM limit.
For the free flow (u f = 0), we have which also leads to eq. (38).
We here conclude that the normalized flow field defined by eq. (35) leads universally to eqs.

Induced Metric
As proposed in Sec. 2, we define the symmetric 2nd-rank tensor field aŝ where µ, ν run from 0 to d, and h is a constant with mass dimension −2. We can interpret this field as the induced metric on the d + 1 dimensional manifold R + × R d from some manifold in R d defined by σ a (z), which classically becomes the N − 1 dimensional sphere. Using eq. (38), the VEV ofĝ µν is given by where g τ τ (τ ) = hτ 2 16 Explicitly ζ 0 (t) is evaluated as See appendix B for the definition and properties of the incomplete gamma function Γ(a, x).

The metric in the IR limit
In the IR limit mτ 1, we have so that This result shows that g µν , the VEV of the induced metricĝ µν , describes a Euclidean AdS space in d + 1 dimensions at mτ 1 for any d. Indeed, with a new variable u = d/2τ , the world line element is expressed as The condition that ds 2 > 0 with the Euclidean signature requires h > 0.

The metric in the UV limit
In the opposite limit mτ 1, on the other hand, we obtain, which leads to The VEV of the metric describes a Euclidean AdS space at d ≥ 3, while a log correction appears at d = 2.

Einstein Tensor
In this section, we consider the VEV of the Einstein tensor G µν . As mentioned in Sec. 2, quantum corrections can be neglected in the large N limit as Therefore, the VEV of the Einstein tensor G µν can be calculated from the VEV of the induced metric, g µν = ĝ µν , in this limit.

Einstein tensor from g µν
If the metric has the following simple form then the Einstein tensor G µν = R µν − Rg µν /2 becomes where From eq. (42), we have After a little algebra, we obtain so that G µν = 0 at d = 1.

IR behaviors
For mτ 1, we have Using the above results, we obtain where the cosmological constant is given by Λ = −(d − 1)/h. Therefore the space defined by the metric g µν is the Euclidean AdS, except for d = 1 where the space is flat (Λ = 0). This is consistent with the result in the previous section.

UV limit
We next consider the behavior of G µν in the UV limit (mτ 1).
These results are also consistent with results in the previous section.

Behavior of the Einstein tensor from UV to IR
If we define the radius of the AdS space R as R 2 = −1/Λ, we have at d ≥ 3. The radius increases toward the infrared.
We show this behavior more explicitly at d = 3 and 4, using the formula given in eqs. (60) and (61) . In Fig. 1, hΛ

Summary
In this paper, we apply the method in Ref. [2]  The induced metric, and thus the geometry, from the flow field with the NLSM normalization, depends only on the renormalized mass m in the original d dimensional ϕ 4 theory, but neither bare parameters (µ 2 , u) separately nor flow parameters (µ 2 f , u f ) at all. This uniqueness of the induced geometry from the NLSM flow may be natural, since the O(N ) ϕ 4 theory becomes a free massive field theory in the large N limit, which does not depend explicitly on the bare coupling constant u including the free and the NLSM limits. In this sense, the large N scalar field theory in d dimensions corresponds to a d + 1 dimensional classical geometry in the large N limit, which becomes the Euclidean AdS in UV and IR limits at d > 2. A posteriori, the NLSM normalization turns out to be an interesting choice to define the RG transformation, as an AdS space emerges in both UV and IR limits.
Since the information about the renormalized coupling constant appears at the NLO in the large N expansion of the d dimensional ϕ 4 model, our next task is to obtain the solution of the flow equation at NLO. Using the solution, we then evaluate quantum corrections to the classical geometry such as the propagation of the induced metricĝ µν on the classical background. It is interesting to see how the information about interactions in the large N field theory appear in quantum corrections to the induced metric. Although it is rather difficult to solve the flow equation at NLO [15], we are currently working on this problem.
In this appendix, we show that an extra divergence appears in the flow field of the λϕ 4 theory, if the bare action is employed for the flow equation, together with perturbation theory in the coupling [12,17]. Although this kind of divergence can be avoided by using the renormalized flow equation such as the free flow equation in [12,18], it is relevant to our flow action, which contains bare parameters of the original theory. We then discuss how this divergence disappears non-perturbatively in the large N limit.

A.1 Renormalization in the original theory
For simplicity, we consider the model at d = 2, 3 defined by where the renormalization is made by At d = 2, 3, we have Hereafter we neglect O(λ 2 R ) contributions. We then have A.2 Divergence in the flowed field We now consider the flow equation, given bẏ at O(λ), where we take µ f = µ and λ f = λ for simplicity. and The two-point function for the flow field is given by where we have The first term is evaluated as = −δ ab δm 2 t 0 dt 1 dk e −(t−t 1 )(k 2 +m 2 )+ik(x−x 1 ) dp e −(t 1 +s)(p 2 +m 2 )+ip(x 1 −y) while the second one is where J 1 (t) ≡ dp e −2t(p 2 +m 2 ) − 1 is UV finite at d = 2, 3.
We finally obtain which shows that t, s dependent renormalization is needed to make the two-point function for the flow field UV finite.