Holography of Linear Dilaton Spacetimes from the Bottom Up

The linear dilaton background is the keystone of a string-derived holographic correspondence beyond AdS$_{d+1}$/CFT$_d$. This motivates an exploration of the $(d+1)$-dimensional linear dilaton spacetime (LD$_{d+1}$) and its holographic properties from the low-energy viewpoint. We first notice that the LD$_{d+1}$ space has simple conformal symmetries, that we use to shape an effective field theory (EFT) on the LD background. We then place a brane in the background to study holography at the level of quantum fields and gravity. We find that the holographic correlators from the EFT feature a pattern of singularities at certain kinematic thresholds. We argue that such singularities can be used to bootstrap the putative $d$-dimensional dual theory using techniques analogous to those of the Cosmological Bootstrap program. Turning on finite temperature, we study the holographic fluid emerging on the brane in the presence of a bulk black hole. We find that the holographic fluid is pressureless for any $d$ due to a cancellation between Weyl curvature and dilaton stress tensor, and verify consistency with the time evolution of the theory. From the fluid thermodynamics, we find a universal temperature and Hagedorn behavior for any $d$. This matches the properties of a CFT$_2$ with large $T\overline T$ deformation, and of little string theory for $d=6$. Both the fluid equation of state and the spectrum of quantum fluctuations suggest that the $d$-dimensional dual theory arising from LD$_{d+1}$ is generically gapped.

The linear dilaton background is the keystone of a string-derived holographic correspondence beyond AdS d+1 /CFT d . This motivates an exploration of the (d+1)-dimensional linear dilaton spacetime (LD d+1 ) and its holographic properties from the low-energy viewpoint. We first notice that the LD d+1 space has simple conformal symmetries, that we use to shape an effective field theory (EFT) on the LD background. We then place a brane in the background to study holography at the level of quantum fields and gravity. We find that the holographic correlators from the EFT feature a pattern of singularities at certain kinematic thresholds. We argue that such singularities can be used to bootstrap the putative d-dimensional dual theory using techniques analogous to those of the Cosmological Boostrap program. Turning on finite temperature, we study the holographic fluid emerging on the brane in the presence of a bulk black hole. We find that the holographic fluid is pressureless for any d due to a cancellation between Weyl curvature and dilaton stress tensor, and verify consistency with the time evolution of the theory. From the fluid thermodynamics, we find a universal temperature and Hagedorn behavior for any d. This matches the properties of a CFT 2 with large T T deformation, and of little string theory for d = 6. We also find that the holographic fluid entropy exactly matches the bulk black hole Bekenstein-Hawking entropy. Both the fluid equation of state and the spectrum of quantum fluctuations suggest that the d-dimensional dual theory arising from LD d+1 is generically gapped.

Introduction
The theory of quantum gravity likely encodes a holographic principle implying that the information content inside any spacetime can be stored on its boundary [1][2][3][4][5]. The correspondence between AdS spacetimes and CFTs realizes very concretely this vision [6][7][8][9]. It further provides the explicit description of the unitary evolution of boundary data.
Since the holographic principle itself applies to any spacetime, we might expect phenomena similar to AdS/CFT to appear in other geometries, including flat and asymptotically flat spaces. The holographic dual theory of flat space remains elusive so far, although progress is being made via the study of celestial amplitudes, see e.g. [10] and references therein. On the other hand, the holography of a specific asymptotically flat spacetime offers encouraging results. It is the so-called (d + 1)-dimensional linear dilaton (LD d+1 ) spacetime.
The linear dilaton spacetime is a negatively curved spacetime which is, in some sense, right in between AdS and Minkowski spacetimes. This fact can be recognized at the level of the entropy of massive black holes, which grows as E , and E respectively for AdS, flat and LD spacetimes. In this work we obtain the latter property for any d via the properties of the LD holographic fluid. The fact that LD space looks sometimes like flat space also appears in a variety of other ways throughout this work.
Below we briefly recall what is known about LD holography and then expose our approach.
LD holography from strings (review). It has long been known that the linear dilaton background is holographic [11]. The string derivation of this fact involves a stack of NS5 branes taken in an appropriate decoupling limit. It leads to a holographic correspondence between a string theory in a seven-dimensional (7D) LD background and a 6D string theory on a stack of NS5 branes with g s → 0 [11]. The latter limit is known as little string theory (LST) [12,13]. It is an interacting theory of noncritical strings which is nonlocal, has no massless graviton, and has Hagedorn density of states at high energy [14,15]. Lower-dimensional versions of this LD 7 /LST 6 duality can be obtained via spatial compactifications.
Even though we know that the dual LST does exist, it is hard to study it because it seemingly has no independent formulation such as a Lagrangian description. However, it has been recently proposed that a 2D compactification of LST with many fundamental strings may be described as a CFT 2 deformed by a specific single-trace T T deformation [16]. As a result, the 3D gravitational description of this deformed CFT 2 is a string theory living on a background that interpolates between AdS 3 and LD 3 . The LD 3 geometry corresponds to the large T T deformation regime. Further evidence for the proposed duality involves the behavior of correlators and of certain symmetries (see [17][18][19][20][21][22][23], and [24] for a review). These progresses are obtained from the top-down via string models analyses.
LD holography from the bottom up. In this work we propose to explore the (d + 1)-dimensional linear dilaton spacetime and its holographic properties from a bottom-up approach, working in the regime of subPlanckian energy scales. In this regime, gravity can be treated classically and effective field theory (EFT) techniques can be applied. 3 There is an AdS/CFT analog to this approach. At the level of EFT, a version of AdS/CFT can essentially be derived by using symmetries and the conformal bootstrap 4 (see e.g. [8,[32][33][34] as points of entry in the literature.) In analogy with the EFT approach to AdS/CFT, our aim in the present paper is to explore some holographic aspects of the LD d+1 background at low energies. The LD background has, of course, less symmetry than the AdS one. Yet we will find that symmetries do play a role. The overall strategy is to place a brane in the LD d+1 background, and to evaluate the correlators and the thermodynamic behavior as seen from this 'braneworld'.
In this work we do not propose explicit dual theories for the LD d+1 spacetimes. While one may keep in mind that, for certain dimensions, the dual boundary theory might be some LST taken in an appropriate low-energy regime, our study remains agnostic of the current string knowledge about the boundary dual theories. Our focus is on, i) independently deriving familiar features from the low-energy viewpoint, and ii) uncovering new properties. We may wonder, for example, if some principle could be identified in order to bootstrap the d-dimensional dual theory.
Outline. In Sec. 2 we compute the linear dilaton background in any dimension, discuss its global properties, and identify symmetries of the LD line element. In Sec. 3 we build an EFT for a scalar field which is compatible with the symmetries of the LD d+1 background. In Sec. 4 we place a d-dimensional brane in LD d+1 and compute brane correlators on either side. In Sec. 5 we allow a black hole solution in the bulk, compute the resulting holographic fluid on the brane and study its thermodynamic properties. Finally, Sec. 6 contains the detailed summary of our results, while App. A contains additional details on the LD solutions with and without the bulk black hole.
Conventions. Throughout this work we use the conventions of Misner-Thorne-Wheeler [35], which include the mostly-plus metric signature sgn(g M N ) = (−, +, . . . , +). Likewise we define the metric determinant √ g ≡ | det g M N |. The number of spacetime dimensions is D = d + 1, we use either D or d depending on the context.

Bulk Action
We consider a D-dimensional spacetime whose coordinates are labeled as x M . The general scalar-gravity action in the Einstein frame is where R is the D-dimensional Ricci scalar, ϕ is the scalar field, V (ϕ) is the scalar potential, and M D is the fundamental D-dimensional Planck scale. The L m Lagrangian contains matter fields, collectively denoted by Φ, living on the spacetime background. The matter Lagrangian can also depend on ϕ.
We assume that the scalar potential depends exponentially on ϕ as where k is a constant with mass dimension 1 andφ is dimensionless. This potential corresponds to a cosmological constant − (D−2) 2 2 M D−2 D k 2 and no potential in the Jordan/string frame. Throughout the paper we refer to the scalar field ϕ as the dilaton.
We split the D coordinates as and assume a warped metric Using this ansatz, the independent field equations for the metric and the scalar are ] has mass dimension 2. There is a third field equation which is redundant, as it can be expressed in terms of these two equations (see Eq. (A.5)).

The Linear Dilaton Spacetime
A canonical solution to the field equations (2.4)-(2.5) is where L is an integration constant with dimension of length. The field equations (2.4)-(2.5) have a total of three integration constants. All the solutions are equivalent to Eq. (2.6) up to coordinate transformations. The complete set of solutions as a function of the integration constants a, c and L is presented in App. A. The Einstein tensor obtained from the ds 2 LD metric is The Ricci scalar is There is a curvature singularity located at r → 0 for D > 2.
Throughout the paper, we sometimes use the conformal coordinates (x µ , z), z = L log L r . The scalar field varies linearly in z in conformal coordinates, which is why the spacetime was dubbed "linear dilaton".

D = 2 dimensions
We briefly comment on the case of D = 2 spacetime. We notice that for D = 2 the solution simply reduces to a flat space metric since R M N and V (ϕ) vanish identically. We can wonder to which region of flat space it maps exactly. To answer this, we go to Cartesian coordinates. For Lorentzian signature ds 2 LD = dr 2 − r 2 L 2 dt 2 , the mapping to coordinates (x, τ ) is given by r = x, t = L tanh −1 ( τ x ). This implies x 2 ≥ τ 2 , therefore we obtain that the metric maps to 2D flat spacetime restricted to spacelike intervals. In other words, in D = 2 the ds 2 LD metric amounts to a two-dimensional version of the Rindler metric. Our focus in this work is on D > 2.

Global features
Singularity. The curvature singularity appearing at r → 0 for the LD solutions can be classified as "good" in the sense of Refs. [36,37]. We will explicitly see the expected physical properties of the good singularity: It repels quantum fluctuations (Sec. 4) and gets censored by a horizon (Sec. 5).
Boundaries. By going to conformal coordinates with z = L log L r , we can see that the LD line element is proportional to ds 2 ∝ dz 2 + η µν dx µ dx ν with (x µ , z) ∈ R 1,d . Thus the conformal structure of the LD spacetime is a null diamond just like Minkowski spacetime (and unlike AdS). 5 Notice that this is in spite of the fact that the scalar curvature and the potential blow up at z = ∞.
Geodesics. Following [38], the timelike geodesics are attracted towards the singularity, and the geodesic distance grows exponentially with the separation L η µν (x µ . In contrast, the null geodesics escape the singularity. In conformal coordinates they behave as straight lines analogously to flat space. 6

Symmetries
What are the symmetries of the line element of the LD spacetime, Eq. (2.6)? The line element has d-dimensional Poincaré isometries along the constant-r slices. It has other symmetries though.
• The line element has a manifest conformal symmetry: the dilatation which gives ds 2 LD → λ 2 ds 2 LD . That is, under dilatation the line element is equivalent to itself up to a constant Weyl transformation.
• The line element has a manifest discrete (Z 2 ) conformal symmetry: the inversion That is, under inversion the line element is equivalent to itself up to a Weyl transformation.
The discrete conformal symmetry of ds 2 LD is expressed here as an inversion in the (x µ , r) coordinates, but of course, the existence of a symmetry is coordinate-independent. For example, in conformal z = L log L r coordinates, this symmetry is a "conformal parity", i.e. the line element is equivalent to itself under z → −z up to a Weyl transformation. The coordinate-independent statement is that the LD line element is conformally invariant under an involution map. In the rest of the paper we mostly use r coordinates and will refer to the conformal Z 2 symmetry of the line element as the "conformal inversion symmetry".

Symmetry as a defining property
Let us study to which extent the peculiar conformal symmetries identified above uniquely define the linear dilaton spacetime. Poincaré invariance imposes the warped ansatz ds 2 = F (r)dr 2 + G(r)η µν dx µ dx ν . Then requiring that dilatation in r leaves the line element conformaly invariant imposes the form ds 2 = F (r)(dr 2 +(r 2 /L 2 )η µν dx µ dx ν ). Such a metric is more general than the LD one. Further requiring that the conformal weight be a constant, i.e. that the line element be homothetic to itself, we obtain the form the scale transformation amounts to translation along the dimension x d+1 ≡ L log r L . In this particular case the dilatation in r becomes an isometry.
The line element ds 2 a has the conformal inversion symmetry for any a. For a ∈ R /{0} , ds 2 a is the one of the LD spacetime, while for a = 0 it reduces to standard parity along the dimension x d+1 ≡ L log r L . One can easily show that it is the unique involution in r of the ds 2 a metric. Finally, one may ask whether the existence of a conformal symmetry under an involution in r restricts by itself the warped ansatz to be LD: The answer is negative. 7 Extra assumptions are needed to reach the LD spacetime by only imposing symmetry under an involution. This happens for example for a line element L 2(b−2) η µν dx µ dx ν , for which conformal invariance under inversion requires b = 2, which yields Eq. (2.13).
In summary, requiring that the warped ansatz be homothetic to itself under dilatation of r gives rise to either the linear dilaton spacetime or the Minkowski spacetime. The inversion symmetry automatically follows. In short, we can say that there is a symmetry which uniquely defines the linear dilaton spacetime. We have checked that imposing only the inversion symmetry gives a weaker constraint.

Effective Field Theory and S-duality
For D ≥ 4 the theory must be considered as a low-energy effective field theory (EFT) describing gravity at subPlanckian scales. The Einstein-Hilbert action is then the leading order term in a curvature expansion of the quantum effective action. The EFT Lagrangian in the gravity sector reads in general where Λ is the scale at which the effects of the UV-completion of quantum gravity show up. From the viewpoint of the low-energy gravity EFT, it is the typical cutoff scale below which the EFT is valid. The O n,i are local operators made of n curvature tensors, e.g.
and their derivatives. Such corrections happen if any state with mass of order Λ is integrated out in the UV completion.

Validity domain of the linear dilaton spacetime
The EFT breaks down when the series in Eq. (2.14) cannot be truncated, i.e. when the R/Λ 2 expansion breaks down. This validity cutoff of the EFT is reached at e.g. high curvatures or small distances. Since in the LD spacetime the curvature depends on the location in the bulk, the EFT cutoff depends on the location r in the bulk. (Notice this is a general feature of EFT in curved space, which happens even for AdS [39,40]).
In the LD metric Eq. (2.6), and using the corresponding curvature tensor Eq. (2.9), we can see that the domain of validity of the EFT is where we have assumed that the a n,i coefficients are O(1).
The bound (2.15) implies that the inversion symmetry is supported on the restricted domain r ∈ [ 1 Λ , ΛL 2 ]. This interval is non empty if the cutoff scale satisfies The inversion symmetry becomes immaterial if this condition is not satisfied.

From S-duality to inversion symmetry
We notice that the inversion (2.12), a conformal symmetry of the metric, transforms the dilaton expectation value as Such a transformation is reminiscent of the S-duality of string theory (see e.g. [41,42] for introductions). The linear dilaton spacetime, at least in certain dimensions, can be thought of as a compactified limit of type IIB string theory, which is self-dual under S-duality [6,43]. In this case the S-duality identifies two solutions with each other, hence defining a Z 2 symmetry. It is thus not surprising to find in our framework two solutions which are identified to each other via an S duality-like transformation.
The string S-duality exchanges weak and strong coupling, g s → 1/g s . In our EFT this notion maps onto the strength of the higher order curvature corrections to gravity, which varies with r. The region of strong coupling is the one toward the singularity, where the EFT of gravity breaks down. The region of weak coupling is the one away from the singularity, where the EFT is valid. The S-duality suggests us that there should be an alternative weakly coupled description beyond the region where EFT validity breaks down, r < 1 Λ . This is, in a sense, what we observe: there is indeed a weakly-coupled region at small r, namely r < ΛL 2 , which corresponds to the original solution transformed under (2.12).
In the following we do not need to refer to S-duality, and we denote the inversion symmetry as S when acting on the fields.

Fields in a Linear Dilaton Background
In this section we consider quantum fields living on the linear dilaton background. These are contained in the matter Lagrangian in Eq. (2.1). The quantum fields inherit the symmetries of the linear dilaton background. We study the effects of these symmetries on the fields and how they constrain physical observables. For concreteness we focus on a real scalar field Φ(x M ) and its correlators.

Symmetries and Free Fields
The fields living in the LD background Eq. (2.6) are constrained by Lorentz invariance along the constant-r slices, thus Φ = Φ(x µ x µ , r). Moreover they should transform as representations, in field space, of the conformal dilatation, Eq. (2.10), and the conformal inversion, Eq. (2.11), symmetries.
The fundamental action of the matter fields on the LD background is denoted as Let us consider a free field with canonical normalization, The corresponding d'Alembertian is defined by

Dilatation
Since the LD background has conformal dilatation invariance in r, any operator O on the LD background should transform under a dilatation transformation D λ as where we refer to ∆ O as the scaling dimension.
How should a free field scale upon dilatation in r of the LD background? The condition is that the action of the free field should be invariant under the dilatation, This condition is needed to ensure that the correlators (i.e. the derivatives of the partition function) transform as representations of dilatation. This would not be the case if the action transformed nontrivially under dilatation. Using the transformation D λ [Φ(r)] = λ −∆ Φ Φ(r), the scaling dimension of the free field must be The result (3.6) is also independently verified via a direct calculation of the 2-point (2-pt) function. As a side note, in our conventions the dimension Eq. (3.6) happens to match the standard mass dimension of fields in flat space.

Inversion
We have seen that the line element of the background is self-equivalent under the inversion operation (2.11) up to a conformal factor. This transformation can be equivalently thought of as a Weyl transformation i.e. a redefinition of the metric field. We may thus expect that operators on the LD background transform under the inversion up to a similar redefinition. The transformed operator S[O] should transform under dilatations, thus the field redefinition should be a power of r, where c ∈ R is a constant. We have that therefore S is an inversion. Requiring that the matter action be invariant under S, we find that the free field transforms as i.e. c = 1 − d for the free field. The following identities hold: The invariance of the action is then verified by using where we used the change of variable r/L → L/r in the second line.

Mass
A mass term is similarly constrained by the dilatation and inversion symmetries. We find where m is constant. Notice the nontrivial scaling in r.
We may also write the mass term (3.13) in a covariant form that distinguishes the metric factor √ g = (r/L) d from the dilaton background. In that view the mass can be understood as a function of the dilaton ϕ, with m(ϕ) ≡ kLeφm. This form is not needed for our EFT analysis of a matter field living on the LD background.
We can also notice that if we use a non-canonical field Φ ≡ r u L uΦ for any u ∈ R, the kinetic term Eq. (3.2) expressed inΦ generates a mass term with precisely the form of Eq. (3.13). This ensures that the structure of the Lagrangian is not spoiled if one uses noncanonical fields -the symmetries always constrain the mass term to a unique monomial in r.
The mass term Eq. (3.13) together with the d'Alembertian form the wave operator (3.14)

Mass Bound
We find that the mass term defined in Eq. (3.13) must satisfy the condition to avoid tachyon-like instability of the theory. This can be, for example, obtained by putting the equation of motion Eq. (3.14) in Schrödinger form, by going to conformal coordinates z, and performing the field redefinition Φ = e − d−1 2L z Ψ. The equation of motion then takes a flat space form, In analogy to flat space, requiring the absence of violation of causality implies Eq. (3.15). The same conclusion can be obtained by inspection of the Feynman propagator in position space, see Eq. (3.28). The negative lower bound in Eq. (3.15) is somewhat similar to the AdS Breitenlhoner-Freedman bound, even though our argument is analogous to a flat space one.

The free 2-pt correlators
It is convenient to work in position-momentum space (p µ , r) to determine the 2-pt functions of the scalar field. We introduce the Fourier transformed 2-pt functions as G(x, x ′ , r, r ′ ) = d d p (2π) d G(r, r ′ ; p)e ipµ(x−x ′ ) µ . In position-momentum space the equation of motion of the propagator is given by where the wave operator is given in Eq. (3.14), and we will introduce

Modes
The solutions to D r Φ = 0 are r We can see that the quantity λ = Upon suitable normalization, the oscillating solutions form a set of modes {f + , f − } defined as This set of modes satisfies orthogonality under The set of modes also satisfies a completeness relation obtained by summing over all the allowed values of p 2 , The 1 √ 2γp normalization factor of the modes is key for the completeness relation to work out.

The Wightman propagator
The Wightman propagator is the free non-ordered 2-pt function W (x, [44]. It can be thought of as the building block for the other 2-pt functions. The Wightman propagator can be obtained as the sum of normalizable modes of positive energy. Here we have to sum over both the f + and f − modes, which are inequivalent where θ(x) is the Heaviside function.
We can see that the symmetries of the background are reflected in the form of the Wightman propagator. Firstly, the Wightman propagator has scaling dimension ∆ = 2∆ Φ = d − 1, it is thus consistent with the scaling obtained in Eq. (3.6) by requiring invariance of the matter action under dilatation Eq. (3.5). Second, the Wightman function is invariant under the inversion symmetry: This symmetry is pictured in Fig. 1. Apart from these symmetries, we can verify that the Wightman function is Hermitian, W (r ′ , r) * = W (r, r ′ ), which is always true and serves simply as a sanity check of our calculations.

The Feynman propagator
The Feynman propagator is defined by (3.24) In position-momentum space we find where r < = min(r, r ′ ), r > = max(r, r ′ ). Independently from the above derivation, G also corresponds to the solution to the equation of motion Eq. (3.16). It turns out that the LD propagator can be expressed in terms of a flat space propagator. Going to conformal coordinates for convenience and defining G(r, r ′ ; p) ≡Ĝ(z, z ′ ; p), we findĜ whereĜ flat is the propagator of a scalar in d + 1 dimensions with squared mass Having identifiedĜ flat (z, z ′ ; p; M 2 ) as the integrand of the bottom left term, one can easily verify the relation (3.26). Equivalently, in position space with coordinates (x µ , r) and By considering a timelike interval with endpoints with identical r, we can notice that the propagator would blow up if m 2 < − (d−1) 2 4L 2 , indicating violation of causality. This proves the mass bound Eq. (3.15).

Effective field theory
Having understood the symmetries of the free theory, we can build an interacting theory by following the principles of effective field theory. We will write local operators taking schematically the form L eff ⊃ F m,n (r)∂ 2m Φ n . The dilatation and the inversion symmetries completely constrain the F function.

Bilinear operators
The bilinear operators of the theory can be built using powers of the following D'Alembertian, The quantity □ r Φ(r) has the same scaling dimension as Φ, and furthermore transforms as Φ under the inversion operation: We can therefore write the complete bilinear action as with the self-energy Π expanded as (3.32) The n = 1 term is the kinetic term, with a 1 = 1 for canonical normalization. The n = 0 term is the mass term, with a 0 = m 2 (r) Λ 2 .

Interactions
Interaction terms are similarly constrained by the dilatation and inversion symmetries. For example, for a nonderivative interaction Φ n we find As in the case of the mass term, we could also write this interaction covariantly with a dilaton-dependent coupling γ n (ϕ) ≡ (kLeφ) d+1−n d−1 2 γ n . We can also check that the derivative interaction is invariant. The invariance under inversion is not straightforward. It happens due to cancellations, upon integration by part of a Φ ′ Φ n+1 term which appears when applying the symmetry.

Structure of correlators
Consider a diagram contributing to an l-pt correlator ⟨Φ( In position space, the diagram is built using the above interactions together with the position space Feynman propagator of Eq. (3.28). Most r L powers simplify inside the diagram, the remaining ones combine such that the interior of the diagram amounts to an amplitude in flat space with (x µ , L log( L r )) coordinates (or equivalently (x µ , z) in conformal coordinates). The only remaining r L factors are those associated with the endpoints of the external propagators.
The above facts put together imply that any Feynman diagram of our EFT takes the form of a flat space diagram with each external leg ending at r i multiplied by a r i The emergence of a flat space amplitude is a consequence of the symmetries we have imposed to build the EFT. The M flat d+1 (r i ) correlator is invariant under the dilatation of the r i coordinates (or equivalently under translation in the conformal coordinates z i ). The scaling behavior of M LD d+1 (r i ) is thus simply set by the overall powers of r in the righthand side of (3.35). This ensures that the M LD d+1 (r i ) diagrams scale as l d−1 2 , consistent with the scaling dimension of a product of l free fields (see definition in section 3.1.1).

Linear Dilaton Holography
We turn to the holographic properties of the linear dilaton spacetime. We place a flat brane in the spacetime and study the correlators ending on it. 8 From these, we deduce properties that the putative d-dimensional holographic dual theory, if it exists, should satisfy. We also point out that the properties of the holographic correlators can be explored using techniques analogous to those applied to the wavefunction of the Universe.

Two halves of LD
We assume the existence of a flat brane (i.e. a domain wall) along the r = r b surface. All the fields have boundary conditions on the brane. The brane is described by appending the action to the bulk action defined in (2.1). The brane carries a localized potential V b (ϕ). The induced metric determinant in our coordinates is The solution to the bulk field equations is the same as without a brane, but the parameters of the solution differ. While the brane location r b may in principle evolve in time (see section 5.4), the value ofv b does not change as a function of r b , because it is fixed by a brane-localized potential, i.e. ∂v b ∂r b = 0. This fact implies a constraint on the c parameter, Other equivalent conditions are given in section 5.5. The solution of the field equations with the parameter fixed in Eq. (4.3) is The brane splits the spacetime into two regions, as illustrated in Fig. 2. We refer to the r ∈ [0, r b ] region as LD − , and to the r ∈ [r b , ∞] region as LD + . The two regions are fundamentally different: LD − contains the singularity while LD + does not. Our goal is to study the quantum fields living on the LD ± regions from the viewpoint of the brane.

Preliminary observations
Properties of the QFT differ between regions LD − and LD + . As will be shown below, in the LD − region the spectrum in d-momentum space features isolated discrete modes in addition to a gapped continuum. These isolated modes are absent in region LD + . A qualitative way to understand this phenomenon is that the quantum fluctuations tend to be repelled from the curvature singularity, so that in the LD − region there is a notion of confinement between the brane and the singularity.
Various truncations of the LD background have been considered in the literature [25,26,29,31], sometimes involving two branes. Our analysis provides a simple overview of the regions and makes it clear that isolated modes exist even with a single brane. 9 As will be clear below, the isolated modes of the gravity sector in the LD − region are a massless graviton and a massive scalar, usually called radion [29]. The existence of the massive radion mode ensures that the brane-singularity distance is stable in the one-brane setup.
The existence of the isolated massless graviton in the LD − region means that e.g. one can exploit the D = 5 case to build models featuring Einstein-like 4d gravity at low energy. This phenomenological line has been followed to some extent in [29]. In contrast, in the LD + region there is no graviton massless mode. Hence at energies below the graviton mass gap, gravity can be integrated out. In the LD + region we thus recover the familiar LST property that gravity decouples at low-energy [14,15].
Our focus in this work is not on the isolated modes of the LD − region, but rather on the theory as seen from the brane. That is, from the viewpoint of the partition function, we probe the theory by placing sources on the brane. Integrating out the bulk modes defines the brane quantum effective action, which generates the brane correlators. Unlike an EFT for isolated modes, this holographic viewpoint applies for both LD − and LD + regions. Integrating out either the LD + or LD − region of the bulk gives rise to two inequivalent holographic effective theories described by the quantum effective actions Γ + and Γ − .
Throughout this section we are not interested in the r b dependence of the correlators. We then simplify the expressions by assuming the brane located at For the correlators we compute in this section, assumption (4.5) is equivalent to stating that the correlatorsĜ ≡ G| r b =η −1 are functions of the rescaled coordinates (x µ ,r) witĥ

Fields and Propagators
We study the quantum fluctuations of a matter scalar field Φ and of the bulk metric in the LD ± regions. We assume that the propagators are regular at r → 0 and r → ∞, just like in the LD spacetime without brane. We assume Neumann boundary condition for the matter scalar. For the metric fluctuation our focus is the tensor component g µν , i.e. the d-dimensional graviton, which has automatically Neumann boundary condition on the brane.

Matter Scalar
We consider the matter scalar field Φ living on the LD background in the presence of the brane. The action and properties of this field in the full LD background have been studied in Sec. 3. We denote its bulk action as S bulk Φ . The full action describing the effect of the brane on Φ is The surface term, which contains the function B, contains brane-localized operators which are bilinear in Φ and depend on the flat-space d-dimensional d'Alembertian. Upon Fourier transform along the flat slices, the bulk action gives rise to the bulk wave operator D given in (3.14). The corresponding bulk equation of motion is given in (3.16). The Neumann boundary conditions on the Green's functions of Φ are With this boundary condition, the Feynman propagator G ± Φ in regions LD ± is found to be where B ≡ B(−p 2 ).

Graviton
We define the graviton as the traceless fluctuation of the metric where we have used Eq. (4.5). We introduce the graviton propagator as The reduced graviton equation of motion is then simply expressed with the scalar wave operator with boundary condition ∂ r G h | r=r b = 0 on the brane. The reduced bulk Feynman propagator in the LD ± regions is related to the scalar one. We find . (4.14) Since the graviton propagator can be expressed in terms of the scalar propagator, our focus below is essentially on the scalar sector. We shall drop the Φ indexes from the scalar quantities, while the graviton case is obtained by specializing to m → 0, B → 0.

Holographic Correlators
We now compute the entries of the boundary quantum effective action of the fields, denoted by Γ ± [Φ b ], where the Φ b argument is the classical field value on the brane. The effective actions Γ + and Γ − are obtained from integrating out, respectively, the LD + and LD − regions. They are in principle different. The boundary quantum effective action can be conveniently evaluated by expressing the fields in a holographic basis which separates the bulk and brane degrees of freedom. In momentum space it takes the form Φ(p; is the boundary degree of freedom, and Φ D (p; r) is the Dirichlet component satisfying Φ D (p; r b ) = 0. The K(p; r) function, which satisfies the condition K(p; r b ) = 1 for any p, can be chosen to be the amputated Neumann brane-to-bulk propagator. With this choice the quadratic action is diagonal in the (Φ b , Φ D ) basis [45]. There is a holographic decomposition for each of the LD ± regions, with distinct brane-to-bulk propagators K ± , that are given further below.
The effective action Γ ± [Φ b ] contains the 1PI pieces of the boundary correlators, that can be extracted by taking derivatives in Φ b . Notice that the notion of 1PI is meant with respect to boundary-to-boundary lines. Our focus is the tree-level entries of Γ ± [Φ b ], sometimes referred to as holographic self-energy and vertices.
The tree-level bilinear entry of Γ ± [Φ b ], i.e. the holographic self-energy, is given by .
The last equality is a general property of the holographic decomposition which similarly holds for any background [45].
The higher point terms of Γ[Φ b ], i.e. the holographic vertices, amount to bulk diagrams built from Dirichlet propagators, with the K ± (p j , r j ) in external legs. They take the general form is a l-pt bulk subdiagram made from Dirichlet lines. We use the notation C (l>2) (p i ) ≡ C (l>2) (p 1 , . . . , p l ) and similarly for A and related quantities.

Propagators
The relevant propagagors in the LD ± regions are as follows. The (amputated) brane-tobulk propagators are (4.17) The Dirichlet propagators that appear in internal bulk lines are The holographic basis provides a decomposition of the Neumann propagator into a Dirichlet propagator plus a boundary contribution describing the propagation of the boundary degree of freedom. This Neumann-Dirichlet identity in the LD ± regions is

Dual theory
Finally, we can always interpret Γ[Φ b ] as the generating functional of an unknown ddimensional theory probed by the Φ b source, with O an operator from the dual theory. The C (l) (p i ) are then interpreted as the connected correlators of this putative dual theory. The dual theory is unknown in the sense that we do not know the d-dimensional fundamental action that describes it. Whether this dual description actually exists is a highly nontrivial question. In this work we infer from Γ[Φ b ] some generic features that the dual theory should possess. We emphasize that all our analysis remains relevant independently from the actual existence of the dual theory.

Two-point Correlators and Spectrum
The scalar brane-to-brane propagators in the LD ± regions are From the viewpoint of the brane, these are 2-pt functions in d-dimensional flat space. They thus have a Källén-Lehmann representation from which we can extract a spectral function.
In momentum space the spectral function is given by For convenience we fix the brane operator to be a mass term, i.e. B ≡B/r b whereB is dimensionless. We find with the isolated mode mass given by

The isolated mode
From Eq. (4.23) we see that only the region LD − features an isolated mode. This can be understood as an effect of confinement between the brane and the singularity. This illustrates the idea that the singularity repels the scalar fluctuations. The condition onB from the Heaviside function on the second line in Eq. (4.23) appears because ifB ≥ d−1 2 the residue of the pole vanishes. This mechanism automatically prevents the residue to become negative, i.e. it prevents the presence of a ghost in the spectrum.
We notice that a massless mode can be obtained by tuningB, but only if the bulk mass satisfies m 2 ≥ − (d−1) 2 4L 2 . Interestingly, this is the same bound as the one obtained in section 3.1.3, arising here from a subtler criterion: requiring that a massless isolated mode can exist in the parameter space. 10 The graviton case is obtained by settingB = 0, m = 0. Hence the isolated graviton mode in LD − is massless and the continuous component of the graviton spectrum in LD + and LD − are exactly equal, with a mass gap at (d − 1)/(2L).

Holographic correlators
We turn to the self-energy C (2) ± (p 2 ), defined by Eq. (4.15). This self-energy can always be interpreted as the 2-pt correlator of an unknown d-dimensional dual theory. The spectral distribution of the 2-pt holographic correlator is given by The result from either the LD − or LD + regions is where we remind the definition . We see that the spectral functions are equal. No information on the isolated mode remains in ρ ± C . More generally, the B term entirely drops because it enters as a purely analytical term, which cannot contribute to the spectral distribution.
We can proceed similarly with the graviton field. The brane-to-brane propagator for the graviton in LD ± regions is the same as Eq. (4.21) with m = 0 and B = 0, thus the graviton spectral distribution is ρ ± C (µ) m=0 .

(4.26)
These results show that the putative dual theory emerging from the holography of either LD ± must describe a gapped continuum. A similar conclusion is obtained from other fields, see e.g. [27][28][29][30] for related results. The fact that both regions lead to the same continuum spectrum is nontrivial. It comes from the fact that the 2-pt propagators in LD − and LD + only differ by the isolated mode, which is reminiscent of the inversion symmetry of the LD spacetime. Interestingly enough, a similar spectrum is also obtained from the single-trace deformed CFT 2 studied with string techniques, see Refs. [16][17][18].

Higher-point Correlators and Singularities
We now investigate 3-pt and 4-pt correlators. We use the bulk vertices from the EFT obtained in Sec. 3. These vertices respect the dilatation symmetry by construction.

Contact diagrams
To warm-up we compute a 3-pt contact diagram induced by the cubic vertex The diagram in the LD ± region, for example, is given by We find identical correlators for LD − and LD + , We can see that C (3), con ± has a singularity going as the inverse of the sum of all scaling dimensions, Similarly, the n-pt contact diagrams from a Φ n monomial are identical in both regions, C (n), con + = C (n), con − , and are proportional to 1/∆ T with ∆ T ≡ l i=1 ∆ p i the sum of all the scaling dimensions.

Exchange diagram
We compute an s-channel 4-pt exchange diagram induced by the cubic vertices. We denote the exchanged d-momentum by q µ = (p 1 + p 2 ) µ . It appears in the scaling dimension of the propagator, ∆ q . We define (4.30) We first compute the exchange diagram with an internal Dirichlet propagator. This is, in the language of Sec. 4.3, a contribution to the 4-pt holographic vertex. We find that this exchange diagram is in both regions LD ± .
In order to compute the exchange diagram with a Neumann internal propagator, we can equivalently re-do the calculation using the propagator from Eq. (4.9), or add to Eq. (4.31) the contribution from the boundary term in (4.19). The Neumann exchange diagram takes the form C with the boundary contribution (4.33) The boundary contribution C contribution. We see that the holographic decomposition naturally separates C (4), ex ±,N as a boundary term and a 1PI term. The boundary term can be built from lower-order 1PI holographic vertices. Hence the nontrivial information is rather encoded into the 4-pt 1PI term C (4), ex ± . Interestingly, it is identical in both regions LD ± .
The C (4), ex ± (p i ) term features a singularity at ∆ T → 0. This singularity is reached for physical value of the external momenta. It occurs when the four scaling dimensions simultaneously vanish, The vanishing of a given ∆ p i corresponds to the threshold value −p 2 i = m 2 + (d−1) 2 4L 2 beyond which the 2-pt line (here the K(p i , r i )) develops an imaginary part. This is, physically, the kinematic threshold of the gapped continuum found in Eq. (4.25). The ∆ T = 0 singularity thus occurs at the configuration of simultaneous kinematic thresholds, Eq. (4.34).
Since the ∆ T → 0 singularity has a physical meaning, the associated residue may also have one. We find that the residue of the 1/∆ T pole is We recognize a 4-pt amplitude from d-dimensional flat space. It is, up to a positive overall factor, the S-matrix element for the exchange of a scalar with mass m 2 + (d−1) 2 4L 2 . In the next subsections we explain why these features happen.
Finally we comment on the other singularities of C (4), ex ± (p i ). Singularities associated with the cubic subdiagrams occur if ∆ 1,2 , or ∆ 3,4 , and ∆ q simultaneously go to zero. The threshold in ∆ q requires −q 2 > 0 to be reached, which occurs in case of an s-channel. These subdiagram divergences correspond to those of the 3-pt holographic vertices, Eq. (4.29). They thus appear in both the 1PI term C

Singularities
A pattern emerges upon inspection of the perturbative boundary correlators. The singularities are associated with the scaling dimensions flowing through each vertices of the diagrams. In particular, the correlator always features a singularity at vanishing total scaling dimension ∆ T → 0. Let us prove this last point for a general diagram.
We consider the general structure for the generic brane correlator given in Eq. (4.16), that we reproduce here: (4.36) Here for concreteness we focus on the LD − region and drop the − subscript. The result in the LD + region is identical. We notice that the A (l) subdiagram has similar structure as the bulk correlator described in (3.35), except that the external legs are amputated. More precisely, using the same steps as in Sec. 3

.3.3 we find
where A We have extracted the overall delta function which results from the translation invariance in z coordinates. We introduce Eq. (4.38) in the complete expression of the brane correlator C (l) (p i ), Eq. (4.36), and integrate over the z j coordinates. The outcome, and the subsequent steps leading to the final form, are An easy way to understand the scaling dimension of A (l) LD d+1 is as follows. Start from the bulk correlator (3.35), which has scaling dimension l d−1 2 . Each amputation removes two powers of Φ and one integral in r. This amounts to a shift of the scaling dimension by 1 − d and 1 respectively. Doing this for the l legs amounts to shifting the scaling dimension by +l(2 − d), which reproduces the scaling dimension of A where we use the notation ∆ i ≡ ∆ p i , with the overall factor (4.42) From line (4.39) to (4.40) we integrated the δ function in p z l . From line (4.40) to (4.41) we performed the remaining integrals by closing the contour downward to pick the single poles, p z i = −i∆ i . The pole outside the bracket lies upward in the complex p z plane and is thus not picked when closing the contours. We see that the combination l k=1 ∆ k ≡ ∆ T appears in the denominator. Thus our calculation makes it clear that the presence of the 1/∆ T pole follows from conservation of p z . It is thus a consequence of the translation invariance in z ofÂ (l) flat d+1 (p j , z j ), or equivalently of the invariance under dilatation of A (l) flat d+1 (p j , r j ). Finally, taking the limit of simultaneous kinematic threshold ∆ i → 0, the residue of the ∆ T = 0 pole isÃ This residue is precisely the (on-shell) S-matrix element in d-dimensional space. This explains the residues obtained for the 3-pt and 4-pt diagrams (4.29), (4.35). The N T factor reproduces the overall factors in the residues. 12 Our analysis shows that the existence of the singularity and its peculiar residue are a consequence of the dilatation symmetry of the EFT.

Connection to the Cosmological Bootstrap
The intriguing structure emerging from the brane correlators in the LD background is reminiscent of the one appearing in the wavefunction of the Universe in dS 3 space.
Studies of the wavefunction of the Universe frequently use a flat space toy model with diagrams ending on a constant time hypersurface [47][48][49][50]. A singularity structure is observed in the wavefunction toy model, which is identical to the one obtained in our LD correlators upon substituting the scaling dimensions ∆ i with energies E i . The fact that the residue at ∆ T = 0 is a flat space amplitude was also noticed, see e.g. [47]. The similarity between the LD correlators and the wavefunction toy model comes from the fact that the spacetime backgrounds share similar symmetries.
To exhibit the connection between these seemingly very different backgrounds, let us first discuss the case of a brane in flat space. We have seen that the dilatation symmetry constrains the form of the brane correlators. Taking L → ∞, LD d+1 space becomes the half flat space with a boundary at z = 0. In this limit the dilatation symmetry reduces to the one-dimensional translation symmetry in the z coordinate (see Sec. 2.3.1). This translation symmetry constrains the flat space correlators, just like dilatation symmetry constrains the LD correlators. In the L → ∞ limit, ∆ i /L tends to the momentum transverse to the brane, p z i . Accordingly, ∆ T /L of a given diagram reduces to the sum of the external p z momenta.
The wavefunction toy model is the spacelike version of a brane in flat space. The coordinate transverse to the hypersurface is time, and accordingly the conjugate variable is energy. The correlators are analogous to those for a brane in flat spacetime upon substituting p z i → E i . This is why the singularities in this model involve sums of energies, i E i , instead of i p z i as for a brane in flat spacetime, and i ∆ i for a brane in LD spacetime. The connection between the LD brane correlators and the wavefunction coefficients is especially interesting because techniques have been developed to study the latter via e.g. analytical bootstrap [50][51][52] and polytope representations [47,48,52]. In particular, the singularity structure (analogous to the one of the LD correlators) essentially fixes the wavefunction coefficients, and tools are developed to further bootstrap the wavefunction of the Universe [50]. It is thus natural to expect that the LD brane correlators can analogously be constrained using their singularities.
An important difference between the wavefunction coefficients and our LD correlators is that, in the former, the singularities appear only upon analytical continuation, while the singularities that appear in our case are physical. 13 Thus it could make sense to take these singularities as input and use them to bootstrap the holographic dual theory. These developments are left for future investigation.

Holography at Finite Temperature
We study the gravitational and thermodynamic behavior of the d-dimensional holographic theory in the presence of a planar bulk black hole.

The Linear Dilaton Black Hole
The general scalar-gravity action in D dimensions in the presence of a brane at r = r b is defined in section 4, see Fig. 2. Here we solve the field equations allowing for the existence of a black hole in the metric,  There is a fourth field equation which is redundant, as it can be expressed in terms of these three equations (see App. A.2 for a discussion). This system admits the following solution where the scale η is defined by η ≡ k ev b . (5.8) As in the zero temperature case, the c parameter is fixed to c = 1 ηr b because ∂v b ∂r b = 0 (see section 4.1). This constraint implies for instance the fundamental property that the d-dimensional Planck scale is independent of r b , ensuring the consistency of gravity in the d-dimensional theory. See Sec. 5.5 for other implications.
The curvature singularity at r = 0 gets censored by the black hole horizon at r h > 0, as expected from a good singularity. Moreover, assuming the EFT of gravity breaks down at distances ∆x ∼ 1 Λ , we require In turn, the horizon censors the region where the EFT of gravity would break down. The geometry is summarized in Fig. 3. The fundamental domain of the r coordinate is r ∈ [0, r b ], so that, in the EFT r ∈ [Λ −1 , r b ], as shown in Fig. 3. For convenience, throughout this section we use a Z 2 orbifold convention as in [56] which implies that the spacetime is mirrored on the other side of the brane. The domain of integration of r is thus doubled in the action. A notable implication is that the entropy of the black hole horizon is doubled when using this convention, see e.g. Ref. [53].
Black hole thermodynamics. Here we summarize some results from the thermodynamic of the black hole. For more details about the calculation, see App. A.2.
The Hawking temperature of the black horizon is The entropy of the black hole may be computed by using a Bekenstein-Hawking-type entropy formula 14 is the D-dimensional Newton constant. We introduce the entropy density per unit of comoving volume, Using the metric solution Eq. (5.5), we find (5.12) Notice that the entropy density depends only on the horizon position r h while the temperature T h is a function of the brane position r b . Therefore the black hole entropy and temperature are independent of each other. This is a special property of the LD background, which does not occur in general in other scalar-gravity systems (see [57] for related results in D = 5).

The Effective Einstein Equations
We now turn to the gravitational behavior of the theory projected onto the brane. The induced metric on the brane at r = r b is where the brane proper time is dt = e −A(r b ) f (r b ) dτ . Notice the e −A(r b ) amounts to a spatial scale factor. The metric (5.13) can be understood as a generalization of the FRW metric to arbitrary dimension.
To study gravity in the holographic theory, we compute the effective d-dimensional Einstein equations as seen by a brane observer. These are computed from the D-dimensional Einstein equations by projecting on the brane via the Gauss equation and the Israel junction condition (see [56] for the original calculation in AdS 5 ). We recall we use the same orbifold convention as in [56].
We find that the effective Einstein equations have the form where T b µν is the stress tensor of possible brane-localized matter. The indices in (5.14) are contracted with the induced metric (5.13). Equation (5.14) has the form of the standard Einstein equations with an extra effective stress tensor T eff µν . Moreover, it turns out that the structure of the T eff µν tensor represents a d-dimensional perfect fluid at rest: 15 T eff,µ ν = g µλ T eff λν = diag(−ρ, P, · · · , P ) . (5.16) In our notation, the effective d-dimensional cosmological constant is included in T eff µν . We work in the low energy regime such that the higher-order terms in Eq. (5.14) can be neglected. This implies ρ b ≪ M d−2 d η 2 , so that ρ b is negligible as compared to ρ eff .

Holographic fluid from warped spacetime
We present the result for the effective stress tensor obtained from the general metric (5.1). The effective stress tensor contains three contributions which are as follows. where n M is the unit vector normal to the brane, i.e.
n M = 0, · · · , 0, f (r)e B(r) , (5.20) andḡ M N = g M N − n M n N is the induced metric on the brane. This leads to the following values for the energy density ρ W and pressure P W : The stress-energy tensor of a perfect fluid is in general expressed as [58] Tµν = (ρ + P )UµUν + P gµν , (5.15) where ρ is the energy density, P is the pressure, while U µ = γ(1, v i ) is the local fluid velocity with γ = 1/ √ 1 − ⃗ v 2 the Lorentz factor. The perfect fluid at rest, i.e. in a comoving reference frame, has U µ = (1, ⃗ 0). The fact that we obtain a fluid at rest is tied to our choice of a static black hole ansatz in the metric. Boosted black hole solutions are also considered in the context of the fluid/gravity correspondence [59] in order to compute higher order corrections to the constitutive relations of the fluid, which involve viscosities and other transport coefficients.
(ii) The τ ϕ µν is from the projection of the bulk stress tensor, (5.32) It turns out that the first term of ρ Λ cancels the constant contribution in ρ ϕ (r b ) in the sum (5.27), so that the effective energy density is given by On the other hand, we find the following results for the pressure, so that the summation of all the contributions yields This result can be interpreted as a pressureless perfect fluid living in a d-dimensional spacetime with a non-vanishing vacuum energy density Λ d , i.e.
In summary, the LD geometry induces a holographic theory which contains a pressureless fluid. This is in contrast with AdS d+1 geometry, where carrying out an analogous calculation leads to a holographic fluid with nonzero pressure P AdS fluid = 1 d−1 ρ AdS fluid , in accordance with d-dimensional conformal invariance. 16 In the following subsections we study the consistency of these results, as well as some of their physical consequences.

Thermodynamics of the Holographic Fluid
A thermodynamic description of the d-dimensional theory can be inferred from the effective energy density derived in Eq. (5.33). The physical parameters of the system are the horizon location r h and the brane location r b . We introduce thermodynamical variables which are understood as functions of these parameters.
First we define the volume and temperature of the system. The spatial volume of the brane is given by x is the brane comoving volume. The black hole temperature on the brane is where T h is the horizon temperature (see Eq. (A.46)). The temperature on the brane is obtained from the horizon temperature by multiplying it by 1/ |g τ τ | [60]. It turns out that T b is independent of r b , and thus from V b . 17 Moreover T b is also independent of r h , it is thus a universal constant of the d-dimensional theory. This is a special property of the LD background. We will see later that T b can be identified with the Hagedorn temperature. By the fundamental laws of thermodynamics, the reversible variation of total energy E of the system should satisfy dE = T dS − P dV . (5.42) Using that E, S and V are 0-forms and d(dX) = 0 for any k-form X, taking the exterior derivative of (5.42) gives dT ∧ dS − dP ∧ dV = 0, where ∧ denotes the exterior product.
Considering that: i) T b is independent on r h and r b which implies dT b = 0, and ii) V b depends only on r b ; it follows that P fluid depends at most on r b . Finally P fluid (r b ) should vanish when r h → 0 since the black hole ceases to exist in that limit, hence P fluid = 0. We thus recover from thermodynamics the vanishing pressure that we found in Sec. 5.2.2. This provides a consistency check of our thermodynamic approach. On the other hand, using the energy density ρ fluid found in (5.38), it turns out that E fluid = ρ fluid V b is independent of r b . Therefore, we get from the relation (5.42) where we have kept for the moment the contribution of the pressure. Notice that each bracket must separately vanish. From the second bracket in Eq. (5.43) we see that the condition P fluid = 0 implies that S fluid is independent of r b . Moreover, from the first bracket in Eq. (5.43) we similarly infer that E fluid and S fluid are proportional to each other, up to an additive integration constant. We set the additive constant to zero using that S fluid and E fluid should both vanish in the r h → 0 limit, for which the black hole does not exist.
In sum, we obtain from (5.43) the relation This automatically implies that the fluid free energy F = E − T S vanishes, F fluid = 0. 18 17 In the present analysis we are assuming r h ≪ r b , so that the r b dependence of T b induced by the 1/ f (r b ) factor is a small correction that we neglect. 18 An alternative proof of Eq. (5.44) can be done by using dF = −SdT − P dV . Taking the exterior Finally, the total entropy is deduced from Eq. (5.44) by using the value of the temperature in (5.41). The total energy and entropy are We notice that S fluid can be independently derived from the Bekenstein-Hawking entropy of the black hole, Eq. (5.11). Starting from Eq.
where we have used the expression of V b given by Eq. (5.40). From a comparison with Eqs. (5.11)-(5.12) one easily realizes that S b = S h , which is a consequence of the independence of S b on r b . Hence the entropy of the holographic fluid matches exactly the black hole entropy, S fluid = S h . Summarizing, we have found that the holographic fluid has a universal temperature, a vanishing pressure and has S fluid ∝ E fluid . These peculiar features reproduce the Hagedorn behavior [61] that typically appears in string thermodynamics [62,63] and is, in particular, obtained for LST [64][65][66][67]. 19 This behavior also occurs for CFT 2 with T T deformations [16]. Here the same phenomenon arises in our low-energy approach, from the projection of the bulk physics onto the brane. The Hagedorn behavior appears generically for any d > 2.

Time Evolution of the Holographic Fluid
Further nontrivial checks of the fluid properties obtained in Secs. 5.2.2 and 5.3 can be achieved by studying the time evolution of the brane. This is, in a sense, a "cosmological" study of the d-dimensional linear dilaton braneworld.
We allow the brane to evolve with time in the bulk: r b = r b (t). The effective Einstein equations Eq. (5.14) give the first d-dimensional Friedmann equation on the brane: where H ≡ȧ b (t)/a b (t) is the Hubble parameter with a b = e −A(r b ) , and ρ b is the energy density of brane localized matter which is neglected as ρ b ≪ ρ eff . When using the black derivative of this relation and using similar arguments, one concludes that P fluid = 0 and then dF fluid = 0. Hence F fluid is a constant in r h and r b that should vanish, and so E fluid = T b S fluid . This proof uses only that T b is independent on r h and r b , and V b = V b (r b ). 19 Identifying the brane temperature T b , Eq. (5.41), with the Hagedorn temperature TH from LST [66] given by TH = (2π √ α ′ N ) −1 , we obtain η = 5/ √ 4α ′ N . α ′ is the string tension, N is the number of NS5-branes, and we have set d = 6.
It turns out that: i) a radiation dominated universe (w = 1 , and iii) a universe dominated by the cosmological constant (w = −1) behaves as log a b (t) ∝ t with ρ ∝ cte. A comparison with Eq. (5.38) confirms that the energy density ρ fluid behaves as a pressureless matter term in the d-dimensional Friedmann equation. 20

Conservation equation
The conservation equation of the D-dimensional bulk stress tensor (i.e. the Bianchi identity) evaluated on the brane can be written as [68,69]  Using the explicit expression of T ϕ M N and after some non-trivial cancellations, it turns out that T ϕ M N u M n M = 0 in the low-energy regime. Finally, by using that it is easy to verify that the conservation equation is satisfied by the effective energy density ρ eff of Eqs. (5.37)-(5.39). A similar analysis to that performed above was done in Refs. [54,55] for d = 4 and ρ vacuum = 0. The present results agree and generalize those of the latter references. 20 For completeness, we provide the solution of the Friedmann equation including both the perfect fluid contribution and the cosmological constant term: Finally, the conservation of entropy is automatically ensured since S fluid is independent of r b , and thus from the brane proper time.

Remarks on the General Solution
We discuss in more details the fixing of the parameter c appearing in the general solution of the field equations with no black hole (A.10)-(A.11), and in the presence of a planar black hole (A.35)-(A.38). We find that c is fixed by various equivalent conditions.
We first impose the bulk conservation equation Eq. (5.51) (i.e. the bulk Bianchi identity) and work in the low-energy regime as defined by Eq. (5.17). Then the following conditions are equivalent to each other: i) The dilaton vev on the branev b ≡φ(r b ) is independent of r b , and thus does not vary iv) Likewise for the effective pressure; dP eff dr b = 0.
vi) The energy loss from the brane into the bulk is negligible; We can unambiguously fix the parameter c by using the general solution forφ(r), together with the definitionsv b ≡φ(r b ) and η ≡ k ev b . The result is and η is independent of r b as is clear from condition i).
Notice that most of the above conditions are physically meaningful. Their equivalence is a compelling verification of the self-consistency of our results. The The power corresponds to w = 0, in agreement with the pressureless fluid result. As a final remark, we mention that the parameter a remains unconstrained. It can thus be freely chosen and the geometry does not depend on it, in accordance with the discussion in App. A. This property becomes explicit, for instance, in the solution of the Friedmann equation provided in Eqs. (5.50) and (5.55), which are independent of a.

Summary and Outlook
In this work we have studied the (d + 1)-dimensional linear dilaton (LD) spacetime with a focus on holography. Let us recapitulate our results.
The LD spacetime has a (good) curvature singularity but no boundary. Its conformal structure is the same as Minkowski space. The timelike geodesics are attracted towards the singularity while the null geodesics behave very much as in flat space.
The LD spacetime has symmetries besides the d-dimensional Poincaré invariance of the flat slicing: a conformal dilatation symmetry and a conformal inversion symmetry acting on the warped coordinate r. The inversion symmetry is somewhat reminiscent of the string S-duality, which is known to be a self-duality of the string UV completion of LD spacetime. We show that the dilatation symmetry uniquely defines the LD geometry, the exact requirement is that the line element be homothetic to itself under dilatation.
We study quantum fields living on the LD background. Following the principles of effective field theory we construct an interactive, massive QFT on the LD background that respects the dilatation and inversion symmetry. We find that the mass term for a scalar field has a negative lower bound. This bound is similar to the Breitenlohner-Freedman bound from AdS, but the proof is closer in spirit to a flat space argument.
We find that the scalar propagator in LD is related to a (d + 1)-dimensional flat space propagator for a scalar with specific mass, up to r-dependent scaling factors. We analogously find that all the correlators of our EFT have a very peculiar structure, they are proportional to (d + 1)-dimensional flat space correlators with r-dependent scaling factors in the external legs.
In order to study the holography of the LD spacetime, we place a flat brane parallel to the singularity and compute the boundary quantum effective action on it. The brane splits the spacetime into two inequivalent regions, LD ± .
We study the 2-pt functions for a matter scalar and graviton in LD ± . The correlators in each regions are almost identical, except that the spectrum in the LD − region features an isolated mode. The similarity of the LD + and LD − correlators is a manifestation of the inversion symmetry of the background. The existence of the isolated mode in LD − shows that the singularity repels quantum fluctuations, i.e. it acts as a (very smooth) boundary for the fields. The LD + region features no isolated mode, which implies that gravity decouples at low-energy. This reflects the gravity decoupling expected in LST. We find that the 2-pt correlators always feature a gapped continuum for any d. This implies that the putative dual theory living on the brane, defined through the boundary quantum effective action, has a gapped spectrum.
We evaluate simple scalar 3-pt and 4-pt holographic correlators built from the boundary quantum effective action. These correlators share peculiar features that also generalize to more complex diagrams: They develop poles at momentum space configurations corresponding to simultaneous kinematic thresholds. Furthermore the associated residue is proportional to a (d + 1)-dimensional flat space S-matrix amplitude. Focusing on the main singularity, we show that both of these phenomena occur for any diagram, and make it clear that these are consequences of the dilatation symmetry of the EFT.
We point out that the peculiar properties of the LD holographic correlators are reminiscent of those of a flat space toy-model used to understand the singularity structure of the coefficients of the wavefunction of the Universe. The analogy implies that bootstrap techniques developed for the wavefunction of the Universe can be transposed to the holographic correlators of the LD spacetime. An important difference is that singularities of the wavefunction coefficients are unphysical while those of the LD holographic correlators are physical. Since such singularities essentially define the correlators, they may even be used to understand/define the putative d-dimensional dual theory, if it exists. Overall, the perturbative correlators seem to tell us that the holographic theory is gapped. This idea matches the result for single-trace T T -deformed CFT 2 derived via string techniques in Refs. [16][17][18].
We then take a completely different viewpoint on LD holography by putting the spacetime at finite temperature. Namely, we solve the (d + 1)-dimensional field equations in the presence of a planar black hole (i.e. black brane). We then project the bulk physics onto the brane, which generates the d-dimensional effective Einstein equation with a nontrivial stress tensor. The effective stress tensor is the manifestation of the bulk horizon on the brane. It receives contributions from both the dilaton bulk stress tensor and from the Weyl curvature projected onto the brane. The pressure from both terms cancel, such that the effective stress tensor matches the one of a pressureless perfect fluid at rest. Summarized: We test this remarkable result by letting the brane evolve with time in the bulk, hence defining a (d + 1)-dimensional linear dilaton braneworld. The evolution of the stress tensor is consistent with pressureless matter. We further check that the (d+1)-dimensional conservation equation projected onto the brane is satisfied. The emergence of a pressureless fluid is reminiscent of a gapped spectrum -since it is the expected behavior for a fluid of massive, nonrelativistic matter. We thus obtain another hint that the dual d-dimensional theory, if it exists, is gapped. This hint, being at the level of the classical gravity solution, is completely independent of our study of the holographic correlators.
We study the thermodynamics of the holographic fluid. We find that E fluid ∝ S fluid and that the proportionality constant is a universal (Hagedorn) temperature. These results match the Hagedorn behavior found in LST and in CFT 2 at large T T deformation. The Hagedorn behavior appears for any d in our low-energy approach. Moreover, we find that the entropy S fluid obtained from brane thermodynamics exactly reproduces the Bekenstein-Hawking entropy of the planar black hole. Summarized: We close by pointing out a few open questions and directions. The computation of the holographic fluid features beautiful cancellations that lead to the vanishing pressure. This vanishing pressure is, to some extent, tied to the conformal dilatation symmetry of the background. It would be good to explore in details the interplay between the dilatation symmetry and the properties of the holographic fluid.
The LD holographic correlators deserve more study. In particular it would be good to analyze them via the bootstrap techniques developed for the wavefunction of the Universe. It would also be interesting to see if the LD correlators admit the polytope representation developed in [47,48,52], analogously to the flat space wavefunction toy-model.
Here we have studied holography on a brane parallel to the singularity, i.e. on a timelike surface. Even though we obtain compelling results, another interesting direction could be to study holography on the conformal null boundary of LD, similarly to flat space holography. This is a distinct analysis that we leave for future work.
Finally the deep, overarching question is whether there is an independent formulation of the d-dimensional dual theory that reproduces the holographic results obtained in this work. Is there, in analogy with AdS d+1 /CFT d , an explicit dual to LD d+1 ? While the existence of LST for d = 6 and the CFT 2 models may be encouraging signals, the mysteries of holography beyond AdS mostly remain to be pierced.
The B(r) function can be freely chosen due to the freedom of redefinition in r. For instance, B(r) can be absorbed by the change of variable e −B(r) dr = dr, which corresponds to proper coordinates. Alternatively, the change of variable e −B(r) dr = e −A(r) dr leads to conformal coordinates. Here, however, we need to keep arbitrary B(r) to find consistent solutions when solving in the presence of a brane.
The field equations are given by (2.4)-(2.5) and a third equation. The three field equations are ] is the solution of Eq. (A.7). These constitute three equations of first order, thus making it clear that there are three integration constants, a feature that has already been discussed in the literature [70].

A.1.2 Integration constants and singularity
Given the potentialV (φ) of Eq. (2.2), the solution of Eq. (A.7) is expressed as where F(x) is the solution of a first order differential equation. It thus contains an integration constant that we will denote by s. By using the method explained in Refs. [53,71], this equation can be solved by making an expansion in s. This produces the following leading term In the following we assume thatV (φ)φ →∞ ∼W 2 (φ) which corresponds to a good singularity (see e.g. Refs. [37,72] for a discussion), and set the integration constant to s = 0. This removes one integration constant from the solutions, leaving only two.

A.1.3 Solutions
A convenient way to derive the general solution of the system (A.2)-(A.4) is to assume that B(r) is linearly related to A(r), i.e. B(r) = α 1 A(r) + α 2 . Then, after setting s = 0 the system admits the following solution, where we have defined the parameters a and c as α 1 ≡ (a − 1)/a and α 2 ≡ − log c. These solutions depend on a set of four parameters: two integration constants (L, r 0 ) and two constants (a, c); which is certainly reducible. The domains of these parameters are a ∈ R /{0} and L, c > 0. Notice that the integration constant L is equivalent to an additive constant c A in A(r), as it can be seen by expressing it as L = e c A /k in addition to c = e ac Ac . In fact fixing c A = −v b leads to the fixing 1/L = η ≡ k ev b , which is usually assumed in warped models [29]. Among the four parameters, r 0 , a and c are redundancies of the description of the D-dimensional geometry, because a change in the value of either of them is equivalent to a diffeormorphism. First, a change in the value of r 0 corresponds to a shift in the coordinate r. We thus set r 0 = 0 without loss of generality. Second, considering the resulting line element the change of coordinates from (x µ a , r a ) to (x µ b , r b ) with a, b ̸ = 0, gives whereφ a (r a ) ≡ −a log gives ds 2 = ds 2 andφ(r) =φ(r). Therefore the map Eq. (A.16) is equivalent to a change in the parameter c. In summary, a and c parametrize redundancies in the decription of the geometry and can thus be fixed. It follows that only one combination of parameters may have a physical meaning.
In order to figure out which combination of parameters exactly is physical, it is instructive to compute the mass gap of the continuum. We focus on the graviton. Using the conformal coordinates z = cL log L r , we can determine the mass gap from the effective Schrödinger potential, The m g parameter is identified as the mass gap. We have thus We can see that the physical quantity m g is related to a combination of the three parameters. In sections 3 and 4 our "gauge-fixing" is a = 1, c = 1, in which case the L parameter is physical, with m g = d−1 2L .
Comparison to Ref. [29]. At this point it would be useful to make connection with the notation used in Ref. [29] for the case D = 5. Using the metric ansatz of [29] ds 2 = e −2A(z) η µν dx µ dx ν + dz 2 , (A. 19) with z = cL log L r , the solution is found to be A(z) = σz ,φ(z) = σz + log |σ| k (A. 20) with the scale σ defined in Eq. (A.18). Here σ can have either sign, and the curvature singularity is in the limit z → ±∞ for a ≷ 0. 21

A.1.4 Fixing the constants
The remaining parameters are fixed as follows. 21 We could also have defined the conformal coordinate z and scale σ as z = sgn(a)cL log L r , σ ≡ |a| cL . (A.21) With this definition, σ is always positive and the singularity is in the limit z → +∞.

i) No brane:
We set a = 1, c = 1 with no loss of generality. The solution turns out to be ds 2 = dr 2 + r 2 L 2 η µν dx µ dx ν ,φ(r) = − log (kr) , (A. 22) and the scale σ is σ = 1/L. Any other value of a, c can be recovered by a coordinate transformation.
ii) Brane at r = r b : Fixing a = 1, the fact that the value of the scalar field at the brane is independent on r b implies (see Sec. 5.5 for other equivalent conditions) The solution turns out to be where the r b -independent scale η is given by The scale σ turns out to be σ = 1 cL = η r b L . (A.26)

A.2 Planar Black Hole solution
Here we will solve the field equations of the metric in the presence of a blackening factor.

A.2.1 Field equations
The metric in the presence of a planar black hole is Then, there are three independent differential equations left, 22 with five integration constants As in the solution with no black hole, this system of equations can also be written as an equivalent system in terms of a superpotential at finite temperature,W (φ) [73]. This is given by 34) in addition to the two equations in (A.6). The system is formed by three differential equations of first order, and one differential equation of second order, thus leading to five integration constants.

A.2.2 Solutions
We follow a similar analysis to that in Secs. A.1.2 and A.1.3. First, we demand a good singularity and set s = 0 which removes one integration constant, leaving four. We find that the system admits the solution The solution has a set of six parameters: four integration constants (L, r 0 , r h , c f ) and two constants (a, c). As in the zero temperature case, r 0 , a, and c are redundancies in the description of the geometry that can be gauge-fixed. We readily set r 0 = 0. The change in a leading to ds 2 = ds 2 , and can thus be fixed. We will consider the value c f = 1 to guarantee that the black hole solution tends to the metric of Eq. (A.1) in the r h → 0 limit.

A.2.3 Temperature of a planar black hole
The temperature of a planar horizon can be obtained by the standard method of demanding the absence of conical singularity. ) 4r h f ′ (r h ) dξ 2 + dx 2 D−2 ∝ dξ 2 + ξ 2 dθ 2 + · · · , (A.45) where we have defined θ ≡ 1 2 e B(r h )−A(r h ) |f ′ (r h )|τ E . This is the usual expression of the line element in polar coordinates. The conical singularity is absent if the line element has periodicity θ → θ + 2π. This translates into a periodicity in Euclidean time of the form τ E → τ E + β with β ≡ 1/T h . We finally arrive at the expression of the temperature This expression is compatible with previous results reported in the literature (see e.g. Ref. [53] for a metric with B(r) = 0).
On the other hand, the entropy of the black hole can be computed by using the Bekenstein-Hawking entropy formula is the D-dimensional Newton constant, and we have introduced an extra factor of 2 from the Z 2 orbifold convention adopted in Sec. 5. By using the metric of Eq. (5.1), the area of the event horizon is computed as .

(A.49)
The entropy density is then given by . (A.50) The zero temperature solution is recovered by taking the limit r a h → 0, which means moving the horizon towards the singularity.
where η is given by Eq. (A.25). The temperature is then where σ is given by Eq. (A.26).