One-loop partition function of gravity with leaky boundary conditions

Leaky boundary conditions in asymptotically AdS spacetimes are relevant to discuss black hole evaporation and the evolution of the Page curve via the island formula. We explore the consequences of leaky boundary conditions on the one-loop partition function of gravity. We focus on JT gravity minimally coupled to a scalar field whose normalizable and non-normalizable modes are both turned on, allowing for leakiness through the AdS boundary. Classically, this yields a flux-balance law relating the scalar news to the time derivative of the mass. Semi-classically, we argue that the usual diffeomorphism-invariant measure is ill-defined, suggesting that the area-non-preserving diffeomorphisms are broken at one loop. We calculate the associated anomaly and its implication on the gravitational Gauss law. Finally, we generalize our arguments to higher dimensions and dS.


Introduction
Large black holes in asymptotically AdS spacetimes are thermodynamically stable if one imposes reflective boundary conditions, such as Dirichlet or Neumann.This is so because they put the black hole in a Hartle-Hawking state since the Hawking radiation bounces off the asymptotic AdS boundary and gets reabsorbed by the black hole, which thus never evaporates.The natural way to evade this boring situation and allow for complete black hole evaporation is to couple AdS spacetime with an external bath that acts as a reservoir for the Hawking radiation (see Figure 1).This construction is the starting set-up for the recent discussions on the Page curve using the island formula in AdS [1][2][3].Remarkably, it was shown in [4,5] that this coupling to the external bath has drastic implications at the quantum level, such as the breaking of the gravitational constraint via a spontaneous symmetry breaking, interpreted as gravitons becoming massive via a Stückelberg mechanism [6].In practice, the coupling with the thermal bath can be described by considering leaky boundary conditions, allowing for the exchange of energy and matter through the permeable boundary.These boundary conditions were studied in detail in [7][8][9] for classical pure general relativity in arbitrary dimensions.The upshot is that the leaks of gravitational waves through the boundary yield nonconservation of the asymptotic charges, which can be described by flux-balance laws (instead of the standard conservation laws associated with reflective boundary conditions).This approach is nonstandard in AdS/CFT where one instead is interested in describing gravity in a closed system (however, for aspects of non-equilibrium AdS/CFT, see e.g.[10][11][12][13][14][15]). Leaky boundary conditions can be considered in other types of asymptotics, such as asymptotically flat spacetimes [16][17][18][19][20][21] where they yield the famous Bondi mass loss formula [22,23], at the future boundary of asymptotically dS spacetime [7,8,[24][25][26][27][28], as well as at boundaries located at finite distance [29][30][31][32][33][34][35][36][37].
In two and three spacetime dimensions, there are no propagating degrees of freedom in pure gravity, which makes the interpretation of leaky boundary conditions less straightforward (see, however, [38][39][40][41][42][43] where they are interpreted in terms of the boundary geometry).Nevertheless, one can implement leaky boundary conditions by simply coupling gravity with a matter field (see, e.g., [44,45] for an example in three-dimensional Einstein-Maxwell theory).In this work, we focus on JT gravity in asymptotically AdS spacetimes minimally coupled to a massless scalar field whose associated propagating degrees of freedom can leak through the conformal boundary.This constitutes the typical setup to discuss the black hole evaporation and the evolution of the Page curve via the island formula.
While this system is perfectly under control at the classical level, we explain why the quantization is a subtle issue, which requires enlarging the Hilbert space to include modes transmitted through the asymptotic boundary.Notably, we observe that the standard diffeomorphism-invariant measure for the scalar field is ill-defined due to the weak falloff of the latter at infinity implied by the inclusion of the extra modes.We explore various alternative quantization schemes to resolve this issue, and are led to the following observation: There is no natural, consistent, local, diffeomorphism-invariant path integral measure for the scalar field.In other words, leaky boundary conditions break parts of the diffeomorphisms in the one-loop partition function of gravity.
We investigate in detail an alternative quantization that involves a Weyl-invariant measure but breaks diffeomorphisms that are not area-preserving [46,47].Hence, the main feature of this quantization scheme is to transfer the usual Weyl anomaly to the non-area-preserving diffeomorphisms.We derive the diffeomorphism anomaly arising at one loop and show that it violates the gravitational constraint.This phenomenon is reminiscent of the higher-dimensional case [4,5] where the gravitational constraint (or gravitational "Gauss law") is broken at the quantum level.Since we are in two dimensions (2d), there is no interpretation of this breaking as bulk gravitons becoming massive.However, this interpretation might be recovered by adding additional degrees of freedom into the theory and re-interpreting the diffeomorphism anomaly as a spontaneous symmetry breaking via the Stückelberg mechanism.
Finally, let us emphasize that the present work takes the point of view of AdS as an open system and does not assume anything about the specific nature of the bath.This constitutes a toy model to address the challenging question of understanding subsystems in gravity [48][49][50][51][52][53][54][55][56][57][58][59].We expect that the closure of the system by adding an explicit bath might shed some light on the exotic features we are discussing here, and we refer to [60] for a recent work along these lines.This work is organized as follows.In section 2, we introduce leaky boundary conditions and derive the classical flux-balance law.In section 3, we study the leaky model semi-classically and point out the issues that lead to the diffeomorphism anomaly.In section 4, we propose employing a local Weylinvariant path integral measure that avoids the difficulties of the previous section.In section 5, we discuss the consequences of leakiness for the Gauss law.In section 6, we put our results into the context of recent literature and generalize the discussion to higher dimensions and dS.In appendix A, we provide a detailed derivation of the asymptotic charges.In appendix B, we give a holographic interpretation of some of our results.In appendix C, we address an alternative dilaton-dependent measure.In appendix D, we provide more details on the Gauss law breaking.

Leaky boundary conditions and flux-balance laws
In this section, we present the main set-up of this paper, namely JT gravity [61,62] (in the linear dilaton sector) minimally coupled to a massless scalar field in AdS 2 with leaky boundary conditions.See [63][64][65][66][67][68][69] for earlier work on (semi-classical) scalar fields coupled to JT gravity.We present a detailed analysis of the asymptotic symmetries in the presence of leaks and explain how the flux-balance laws, which control the non-conservation of the charges, can be deduced from the symmetries.

Solution space
The action functionally depends on the (Lorentzian) metric g µν , the dilaton X, and the matter scalar field ψ.The manifold M is a disk when we continue to Euclidean signature, with boundary ∂M .The gravitational coupling is positive, G > 0, and the other parameter in the action is negative, Λ = − 1 ℓ 2 .The first term on the right-hand side is the bulk JT gravity action, the second is a boundary term ensuring that the variational principle is well-defined for the gravitational sector (see (A.1) or (4.1) of [70] for the explicit expression of L b ), and the third term is a free massless scalar field minimally coupled to the gravitational field.The equations of motion for the action (2.1), guarantee that all solutions are locally AdS 2 .The stress-energy tensor therein, and traceless which is associated, respectively, with classical diffeomorphism-and Weyl-invariance of the scalar field action in (2.1).
We impose the Fefferman-Graham gauge where ρ is the radial coordinate (ρ = 0 at the boundary and ρ > 0 in the bulk) and t the time.We use intrinsic 2d units, where ℓ, t and 1/ρ have length dimension, while X and G are dimensionless. 1 Fixing Dirichlet boundary conditions for the metric, solving the equations of motion (2.2), we find ) where prime denotes derivatives with respect to time t, and we assume X + (t) ̸ = 0 for all times.In appendix B, we give a holographic interpretation of the expansion coefficients above.For completeness, we display the stress-energy tensor falloff, (2.9) As in [39,70], we allow for fluctuations X + (t) ̸ = 0 of the leading order in the dilaton.The quantities J(t) and N (t) correspond to the non-normalizable and normalizable modes of the scalar field ψ, respectively.In addition, the parameter M , which coincides with the leading order of the Casimir function corresponds to the mass, as can be checked on the particular case of JT black hole solutions.In the absence of matter, the latter are given by (2.11) 1 From these choices, we deduce the length dimensions of all other quantities: Tρρ: length squared; N, X − : length; Tρt, J: dimensionless; X + : inverse length; Ttt, C, M : inverse length squared.
The equations of motion (2.2) imply the flux-balance law relating the time derivative of the mass to a bilinear in the scalar field.We need to turn on a timedependent non-normalizable mode J(t) to obtain a non-vanishing flux on the right-hand side.Here, J(t) can be interpreted as a boundary source describing an exchange of energy between AdS 2 and an external bath (see Figure 1).By analogy with the case of asymptotically flat spacetime [16,17,22,23], we call J ′ (t) the scalar news.Once J(t) is turned off or becomes constant, there is no leak through the conformal boundary, and we are back to the standard situation where the mass is conserved in time.
To summarize, the solution space is parametrized by three functions of the time coordinate, X + , N, J, and by the function M whose evolution is constrained by the flux-balance law (2.12).
The non-normalizable modes of the scalar field correspond to those transmitted through the conformal boundary of AdS 2 , while the normalizable modes are those that are reflected.Indeed, the most general solution to the 2d massless Klein-Gordon equation in (2.2) is given by the sum of holomorphic and anti-holomorphic parts, . . .The leading order term corresponds to the non-normalizable mode and the first subleading term to the normalizable mode; further subleading terms are not displayed and do not play a major role.We can also decompose into Fourier modes, f ± = n f ± n e in (t±ρ) .The non-normalizable modes then obey the condition f + n = f − n and the normalizable modes obey the condition f + n = −f − n which are recognized, respectively, as transmissive and reflective boundary conditions on the Fourier modes.

Asymptotic symmetries
Spacetime diffeomorphisms preserving the Fefferman-Graham gauge (2.6) are generated by Here, σ(t) is the parameter for the Penrose-Brown-Henneaux diffeomorphisms [71][72][73] inducing boundary Weyl rescalings, and f (t) is the parameter for the tangential diffeomorphisms leading to reparametrization symmetries at the boundary.Preserving Dirichlet boundary conditions for the metric (2.7) implies σ = f ′ . (2.14) One can easily check that the above diffeomorphisms preserve the falloffs of the dilaton, the metric, and the matter field in (2.8).Assuming no field dependence in the symmetry parameters, the asymptotic symmetry algebra is the standard Diff(R) algebra of reparametrization symmetries at the AdS 2 boundary.
Under the action of infinitesimal symmetry transformations generated by (2.13) with (2.14), the solution space transforms as In appendix B, we give a holographic interpretation of these transformation laws.The identity implies that the zero mode of 1/X + does not transform and, therefore, can be set to some positive value, say, 1/ℓ without constraining the asymptotic symmetries.This condition is required to obtain a well-defined variational principle for pure JT gravity [70].
We conclude this subsection with an important observation: if we want to consider a leaky system where J ′ ̸ = 0 (see the discussion below the flux-balance law (2.12)),keeping the boundary source fixed on the phase space, δJ = 0, would generically break all the reparametrization symmetries at the boundary and enforce f = 0, see (2.16).As we shall argue in the next section, it is crucial to keep the reparametrization symmetries and allow variations of the sources, δJ ̸ = 0, to deduce the flux-balance law (2.12) from the asymptotic charge algebra.

Asymptotic charge algebra
In this subsection, we discuss the variational principle and the charges associated with the asymptotic symmetries derived in the previous section.To do so, we use the covariant phase space methods that allow for a first-principle derivation of these charges [16,74,75] (see, e.g., [76,77] for reviews).The details of the computations of the symplectic structure and the associated charges for the current set-up are presented in detail in appendix A.
As discussed there, the variation of the action (2.1) on solutions is given by a boundary term Reflective boundary conditions on the scalar field require either δJ = 0 = J ′ while keeping N fluctuating on the phase space (Dirichlet boundary conditions) or N = 0 while keeping J fluctuating (Neumann boundary conditions).As their names suggest, these conditions ensure the mass is conserved in time (see the flux-balance law (2.12)).In both cases, the variational principle is well-defined, i.e., δS = 0 on-shell.Indeed, the first term in (2.18) is simply not there 2 .The second term vanishes because the zero mode of 1/X + is assumed to be fixed on the phase space (see the discussion below Equation (2.17)).
In our case, we keep both J and N fluctuating on the phase space to consistently allow for leaks through the AdS boundary.The fact that the variational principle is not stationary on solutions should be expected since we are describing an open system.This is a standard feature appearing when considering leaky boundary conditions [8,9,39].Of course, if one added the bath whose dynamics is described by a variational principle S bath to close the system, we would expect to recover the usual statement δS + δS bath = 0 as a consequence of the cancellation of the symplectic flux between what goes in and what goes out at the AdS boundary.
The expression of the co-dimension 2-form associated with the reparametrization symmetry f is given by Equation (A.14).Using a field-dependent redefinition of the symmetry parameters (sometimes referred to as a change of slicing [31,39,78]), and assuming δ f = 0, we find This expression is a one-form on the phase space, which is integrable (namely δ-exact) in the gravitational sector but remains non-integrable in the matter sector due to the leakiness [16,17].The first term on the right-hand side of (2.20) can be readily integrated to give the gravitational charge In particular, if one sets f = 1, this charge corresponds to the mass of the system.Taking the redefinition (2.19) into account, the symmetry algebra (2.15) is replaced by an abelian algebra, Denoting the non-integrable piece in (2.20) by and using the Barnich-Troessaert bracket, which is adapted for non-integrable expressions [17], we obtain the charge algebra forming a representation of the abelian symmetry algebra (2.22) at the boundary.
From this charge algebra, we can derive a flux-balance law.The boundary-vector generating the reparametrization symmetries can be rewritten as f . In terms of the genuine time, we have the flux-balance law where in the second equality, we used (2.24) and (2.25) with ξ 2 = ξ( f = 1) and ξ 1 = ξ( f ).Therefore, keeping the sources fluctuating on the phase space, δJ ̸ = 0, allows us to recover the flux-balance law controlling the exchanges between the leaky system we are considering and the environment.In particular, we recover the evolution of the mass discussed in the flux-balance law (2.12) as a particular case of (2.26) when setting f = 1.

Leaky boundary conditions lead to diffeomorphism anomaly
In the previous section, we argued that a proper treatment of leaky boundary conditions for the scalar field requires keeping both the normalizable (N ) and non-normalizable (J) modes turned on and as part of the phase space variables.In this section, we quantize this system on a fixed gravitational background using the path integral approach.As we shall argue, the standard diffeomorphism-invariant measure for the scalar field is ill-defined due to the interaction between normalizable and non-normalizable modes.This interaction is already responsible for the leaks, as evident from the right-hand sides of the flux-balance law (2.12) and (2.18).Therefore, all the subtleties discussed in this section are due to leakiness. 3

Problems of the naive approach
The one-loop partition function of the theory (2.1) is obtained by performing the path integral over the scalar field ψ with fixed metric and dilaton background.In the Euclidean partition function the quantity S grav denotes the two first terms of the action (2.1) evaluated on a background solution of the equations of motion.The wavy equality ≈ denotes that we used the saddle point approximation for the gravitational sector of the theory.The remaining factor to compute is the path integral of the scalar field on the fixed background (we drop the bar above the background solution and liberally integrate by parts from now on): In the standard case without exchange between AdS 2 and an external bath, J is fixed and time-independent (usually set to zero).The partition function (3.2) can be computed by using the ultralocal definition of the path integral measure Dψ, , see e.g.[79] on how to evaluate such functional determinants.The measure (3.3) is diffeomorphism invariant but not Weyl-invariant.Whenever a classical symmetry is not respected by the path integral measure it may become anomalous [80].In our context, the quantum theory acquires a Weyl anomaly, leading to a trace in the matter stress-energy tensor: ⟨T µ µ ⟩ = − 1 24π R (see, e.g., [81] for a derivation).
In the leaky case depicted in Figure 1 and described in detail in section 2, the "source" J is turned on at the boundary and is part of the phase space we have to quantize.As a result, the path integral measure (3.3) cannot be used because the falloffs (2.8) imply The two first terms in the expansion diverge, which means that the diffeomorphism-invariant path integral definition (3.3) cannot be taken as it is.Notice that this issue does not arise in the standard case discussed below (3.2) where J is set to 0. In the following, we present various attempts to circumvent this issue.We show that we cannot achieve simultaneously diffeomorphism invariance and locality of the path integral measure.

Introduction of a cutoff
A standard QFT technique of isolating divergences is introducing a cutoff ϵ ≪ 1, which in our case amounts to replacing the integral (3.4) by Employing this regularization, we can attempt to remedy the deficiency of the path integral measure (3.3) by treating the normalizable and non-normalizable modes as two different scalar fields ψ J = J(t) + O(ρ 2 ) and ψ N = ρ N (t) + O(ρ 2 ).We define the regularized path integral measure4 where the matrix M AB according to (3.5) is given by O( 1) . (3.7) Diagonalizing the matrix (3.7) yields the eigenvalues 1) associated with the eigenvectors The cutoff ϵ breaks diffeomorphism invariance, so we need to remove it eventually if we desire a diffeomorphism invariant quantization scheme.If all divergences were monomial, we might rescale our measure suitably by some overall power of ϵ to render the rescaled measure finite, see the discussion in section 5.1 of [60].
However, the second eigenvector in (3.8) depends logarithmically on the cutoff to leading order, so we cannot use this procedure here.Since logarithmic behavior on the cutoff often signifies the presence of an anomaly (e.g., the anomalous leading order divergent contribution to entanglement entropy in a CFT 2 [82,83]), we take the results above as an indication that (at least some of the) diffeomorphisms become anomalous in the leaky quantum theory.For this conclusion, leakiness was essential.Indeed, if we only had the normalizable mode ψ N (or the non-normalizable mode ψ J , but not both simultaneously), we could define a (regularized) diffeomorphism invariant path integral measure (and there would be no logarithmic dependence on the cutoff ϵ).The subtlety highlighted above is due to the interplay between ψ N and ψ J .In the next subsections, we address this fundamental issue from various perspectives.

Non-local measure or diffeomorphism anomaly
As described above, the issue in the path integration arises because of the interplay between normalizable and non-normalizable modes.Hence, a natural way to proceed is to treat normalizable and nonnormalizable modes separately in the path integral, but avoiding the issues with the diagonalization explained in the previous subsection.To do so, we consider a non-normalizable solution ψ J (t, ρ), As a first step, we perform the path integral (3.2) with fixed sources J(t) using the diffeomorphism invariant measure which is perfectly well-defined thanks to the falloffs (3.10).This leads to where h(t) is the boundary metric and is the boundary 2-point correlation function [84].At this stage, the procedure is completely diffeomorphism invariant.In the second step, to capture interactions with the external bath, one has to integrate over the external sources.
Several choices are possible at this stage.If one wants to preserve bulk diffeomorphism invariance, the measure for the sources5 is not local but bi-local.For the free scalar field, this measure makes the sources trivial in the theory.
In the presence of interactions, non-trivial features might appear.We comment on closing the system by adding bath degrees of freedom in the concluding section.
Instead, if one wants to preserve locality in the definition of the measure, then one necessarily breaks parts of the bulk diffeomorphisms.Indeed, the local measure One can easily check that the measure DJ in (3.14) is preserved under the action of diffeomorphisms (2.13) generated by f (t), but not under those generated by σ(t).
Alternatively, the local measure is boundary Weyl-invariant, but not boundary diffeomorphism invariant.Indeed, the radial bulk diffeomorphisms (those generated by σ(t) in (2.13)) preserve the measure (3.15), while the tangent bulk diffeomorphisms (those generated by f (t) in (2.13)) do not preserve the measure.
Since it is impossible to have a local measure that preserves both boundary Weyl and diffeomorphism invariance, locality together with finiteness of the measure imply a breaking of bulk diffeomorphism invariance in the quantum theory at one loop.A loophole in this argument is that we could use the dilaton to make the measure finite, while keeping diffeomorphism invariance.This possibility is pursued in appendix C, but does not lead to a conserved stress-energy tensor either, and moreover does not seem natural for minimally coupled scalar fields.

Local and Weyl-invariant measure
In the previous section, we have seen that locality and bulk diffeomorphism invariance cannot be simultaneously preserved at one loop in the presence of leaky boundary conditions.We now investigate a local path integral measure that is not bulk diffeomorphism invariant but bulk Weyl-invariant [46,47].
is often considered as an alternative of (3.3) and has the advantage of being well-defined in the presence of leaky boundary conditions.This measure does not break all the bulk diffeomorphisms: The areapreserving diffeomorphisms (by which we mean ∂ µ ξ µ = 0) are still symmetries of (4.1).
Before discussing the consequences of using the Weyl-invariant measure (4.1) on the one-loop path integral, let us first review some aspects of the standard diffeomorphism invariant measure (3.3).In the latter case, the effective action is given by the Polyakov action with unity central charge, c = 1, for one massless scalar field, and To render the effective action local, one can introduce an ancillary field6 χ to obtain The χ-equation of motion g χ = 1 2 R can be re-injected into the action (4.3) to recover (4.2).The stress-energy tensor is From this expression, one can now explicitly check that the quantum theory is diffeomorphism invariant but not Weyl-invariant, Let us now repeat the same steps for the Weyl-invariant path integral measure (4.1).In this case, the effective action depends on the Ricci scalar R associated with the rescaled metric γ µν = gµν √ g and γ µν = √ gg µν .Note that R is related to the Ricci scalar associated with As in the standard case, to render the effective action local, one can introduce an ancillary field χ to obtain Again, the χ-equation of motion γ χ = 1 2 R can be re-injected into the action (4.7) to recover (4.6).The stress-energy tensor contains D µ , the covariant derivative with respect to γ µν . 7Setting χ on-shell, we have Since T W µν is a symmetric traceless tensor, we have √ gg µρ ∇ ρ T W µν = γ µρ D ρ T W µν .Comparing (4.9) with (4.5), we see that the anomaly has been moved from the Weyl symmetry to the diffeomorphisms.In other words, the effective action (4.7) is Weyl-invariant, but not diffeomorphism invariant.
The Ricci scalar R constructed from the rescaled metric γ, and hence the anomaly (4.9), depend on the choice of coordinates.For instance, in coordinates where √ g = 1 the anomaly expression ∂ ν R vanishes.However, for self-consistency we should use the set of coordinates (2.6) that defined our boundary conditions.For our specific case, we have where, in the second equality, we have used the notation of appendix B. Thus, the Ricci scalar R to leading order is given by the subleading metric coefficient g tt (2.8b) that contains state-dependent information.
Neglecting backreactions, for a JT black hole background (2.11) we get R = M = const.and hence the anomaly in (4.9) vanishes.However, genuine leaky boundary conditions at the quantum level require including backreactions so that the mass M can actually decrease in time and the black hole can evaporate.In that case, the diffeomorphism anomaly discussed in this section will have a direct physical effect.In the next section, we discuss the impact of the anomaly on the gravitational Gauss law.

Consequences for the Gauss law
As discussed above, if one wants to preserve a local definition for the path integral measure, one has to break part of the bulk diffeomorphisms.The remaining subset of diffeomorphisms that are symmetries of the theory depends on the specific choice of measure.In this section, we discuss the implication of violating diffeomorphism invariance of the theory at one loop when computing the asymptotic charges and derive an explicit violation of the Gauss law.

Classical Noether identities
Let us first review the standard relation between diffeomorphism invariance and Noether identities for the classical theory (2.1).To do so, we use the covariant phase space framework discussed in [75,85] (see also [76,77] for reviews).It is convenient to introduce the following notations: ϕ = (g µν , X, ψ) and Setting the Euler-Lagrange derivatives of L with respect to all fields, (5.1) to zero recovers the equations of motion (2.2).In the identity the Lie derivative, δ ξ ϕ i ≡ L ξ ϕ i , of the field ϕ i with respect to ξ multiplies the Euler-Lagrange derivative of L with respect to ϕ i .The right-hand side of (5.4) is obtained by the Leibniz rule to remove the derivative on the diffeomorphism parameters ξ µ and keeping the total derivative term defining the weakly vanishing Noether current S µ ξ [ϕ] given by (5.5) The second term on the right-hand side of (5.4) exhibits the Noether identities N µ , which identically vanish off-shell as a consequence of diffeomorphism invariance.Indeed, one can check explicitly with the stress-energy tensor (2.3).The on-shell conservation of the stress-energy tensor (2.4) is a direct consequence of the Noether identities.

Quantum gravitational Gauss law
Semi-classically, we consider the theory where L eff [g] is the effective Lagrangian obtained after path integration over the matter field ψ.As discussed in section 4, to make the effective action local and be able to use our methods, we introduce an ancillary field χ in the effective action.The precise form of L eff [g, χ] depends on the particular measure prescription for the integration over the matter field.For instance, if the diffeomorphism invariant measure (3.2) is chosen, then L eff [g, χ] is the Polyakov action (4.3).Alternatively, if (4.1) is chosen, then L eff [g, χ] is the Weyl invariant action (4.7).The expectation value of the stress-energy tensor is obtained from The weakly vanishing Noether current has the same expression as in (5.5) but with L m replaced by L eff .The Noether identities vanish only for quantum theories preserving diffeomorphism invariance.From now on, we focus on the path integral measure (4.1) giving rise to the effective action (4.7) for the matter sector.The violation in the Noether identities can be deduced from (4.8) and reads explicitly (5.9) We continue the analysis keeping this contribution.More precisely, we show that the standard statement that gravitational charges reduce to surface charges (Gauss law ) does not hold if this contribution is non-vanishing.More details about the standard derivation without this contribution can be found in [75,85].The Barnich-Brandt co-dimension 2 form k ξ and the presymplectic form W are defined as ).Here I k δϕ denotes the Anderson homotopy operator acting on k-forms.Using the properties of this operator, we have where ≈ means that we have used the equations of motion by taking into account the backreaction.In Lorentzian signature, integrating over a Cauchy slice Σ, we obtain Surface charge Gauss law breaking . (5.12) This result states that the gravitational charges do not reduce to simple boundary terms.The breaking term on the right-hand side is due to the violation of the Noether identities.In appendix D, we provide a more explicit expression for the breaking term (see Equation (D.24)) and re-derive this result using the Iyer-Wald approach [74], following the steps presented in [42].
The breaking of the standard gravitational Gauss law that we find in (5.12) is reminiscent of what was observed in higher dimensions, where coupling AdS to an external bath breaks the Gauss law and is interpreted as gravitons becoming massive through a Higgs mechanism [4,5].However, as stated in the introduction, this interpretation no longer holds in 2d gravity since there are no propagating degrees of freedom.

Discussion and generalization
In this concluding section, we address implications for black hole evaporation and the addition of a bath system coupled to the black hole.We also comment on some generalizations of the observations raised in this paper.

Implications for black hole evaporation and bath systems
Our starting point was to allow leakiness at the AdS boundary so that (large) black holes in AdS can evaporate.Classically, leakiness implies a non-trivial flux-balance law (2.12) that relates the time derivative of the mass function to (scalar) news.A pivotal technical aspect was that we considered the non-normalizable mode of the scalar as state-dependent, in contrast to a standard AdS/CFT setup where these modes are treated instead as sources that do not vary over the state space.We explained our reasoning for this choice in the last paragraph of subsection 2.2.
Semi-classically, we have seen that the leakiness comes with some quantum baggage: The path integral measure was either ill-defined, non-local, or led to a non-conserved expectation value of the stress-energy tensor.We pursued the last option in detail and showed how the diffeomorphism anomaly (4.9) leads to a breaking of the Gauss law, reminiscent of what happens in higher dimensions [4,5].
These conclusions could potentially be circumvented by adding the bath, which would close our open quantum system.While details of the full system depend on the precise model of the bath, one should always be able to eliminate the diffeomorphism anomaly.For the specific model we studied -JT gravity with a minimally coupled massless scalar field, with both normalizable and non-normalizable modes switched on -such a resolution might be pursued along the lines of [60].The technical key point of that work is a diagonalization of the two scalar modes into a normalizable and a non-normalizable mode, both of which have support on the gravity and the bath sides.By contrast, just working on the AdS side, it was impossible to diagonalize the scalar modes, and the off-diagonal interaction term between normalizable and non-normalizable modes was the origin of the breaking of the Gauss law.We expect that the bath details do not matter for the overall conclusion that adding a bath closes the quantum system and should restore diffeomorphism invariance semi-classically.

Higher dimensions
Let us generalize the discussion of section 3.1 for Einstein gravity in even8 dimension d + 1 (d > 1).We consider gravity coupled to a massless scalar field ψ with the bulk action where Λ = − d(d−1) < 0. The matter sector of the theory is not Weyl-invariant in dimension d > 1 (to restore such an invariance, one would need to add the interaction term d−1 8d ψ 2 R to the action).On-shell, in Fefferman-Graham gauge we have the falloffs where ḡab is a fixed boundary metric, N and J are, respectively, the normalizable and non-normalizable modes of the scalar field.To integrate out the matter field in the path integral, we have to make a choice of measure.The natural diffeomorphism-invariant measure Dψ for the scalar field is defined through If we turn off the non-normalizable J mode, we get which is finite at small ρ.In this case, the usual normalization (6.3) can be used.However, this normalization cannot be achieved if the non-normalizable mode J is turned on as well.Indeed, noticing that √ −g = ρ −(d+1) ℓ √ −ḡ, we have which is divergent and therefore incompatible with (6.3).As in the 2d case, since J is dynamical, it cannot be substracted from the path integral normalization.Therefore, also in higher dimensions one has to use another definition of the measure that will either be non-local or break part of the diffeomorphisms and, consequently, the Gauss law.

de Sitter
So far, we have considered leaky boundary conditions in asymptotically AdS spacetimes, which are suitable to describe interactions between the gravitational system and an external bath.While this set-up is a bit unconventional in AdS, leaky boundary conditions are more common for other types asymptotics.We comment on the possible extension of our results for de Sitter (dS) spacetimes.
In dS spacetime, imposing boundary conditions at the future spacelike boundary I + dS (see Figure 2) is a delicate issue [7,8,24,27,[86][87][88] since it can drastically restrict the Cauchy problem.Indeed, if one prescribes generic data on a Cauchy slice Σ in the bulk, one naturally ends up with leaky boundary conditions at the future spacelike boundary I + dS .Leaky boundary conditions are therefore appealing in this context.Most of the computations performed in this paper could easily be adapted to the dS case, just by changing the sign of the cosmological constant (ℓ 2 → −ℓ 2 ) and considering an expansion near the future boundary I + dS (ρ now becomes a timelike coordinate, while t becomes spacelike).In particular, the problem in the definition of the measure discussed in section 3.1 still applies there and the non-normalizable modes cannot be turned off if one wants to preserve a generic dynamics.It would be interesting to investigate the potential consequences of our work on cosmology and inflation where semi-classical computations play an important role (see [89,90] and references therein).
In conclusion, the subtle interplay between leakiness and 1-loop effects uncovered in this paper is not tied to the specific model we studied in detail -JT gravity with minimally coupled massless matter -but appears to be a rather generic feature of semi-classical gravity with leaky boundary conditions.
Redefining X + = e Φ , the falloff conditions (2.8) for dilaton and metric reduce to9 It is suggestive to interpret the expression L tS as a (twisted) Sugawara stress-energy tensor associated with a current J = Φ ′ of a scalar field Φ.If this interpretation was correct, then expressions like V (h) = e −hΦ should be vertex operators of conformal weight h.
To verify whether this interpretation makes sense, we reconsider the transformation laws (2.16) under reparametrization symmetry at the boundary.The results are compatible with the suggestive CFT interpretation above: Φ is an anomalous scalar field (transforming like a Liouville field or like entanglement entropy, see, e.g., [94][95][96]), J is an anomalous current (transforming with a twist cocycle, see, e.g., [97][98][99]), L tS is an anomalous stress-energy tensor (transforming with the Schwarzian derivative, see, e.g., [72,100,101]), and V (h) transforms like a conformal primary of weight h, i.e., like a vertex operator (see, e.g., [102]).In the main text, we indirectly used the last fact in (2.17) to conclude that the zero mode of 1/X + = V (1) does not transform under reparametrizations.
Finally, to make this appendix self-contained, we list again the transformation properties of some of the remaining quantities, in particular, the mass function M and the modes J and N of the scalar field: Under reparametrization symmetry at the boundary, J and M transform as scalars, N as a weight-1 conformal primary, and e −2Φ M as a weight-2 conformal primary (without anomalous term).Plugging these results into the leading order expressions of the bulk stress-energy tensor (2.9) reveals that each of its leading order components from a boundary perspective is a weight-2 primary (without anomalous term), see (B.2i).The last two expressions show that the flux-balance law (2.12) is consistent with the conformal transformation properties, i.e., the term N J ′ e Φ transforms precisely as M ′ does.
Thus, from a chiral CFT 2 perspective, the subleading term in the metric component g tt (B.1b) is the sum of a weight-2 primary, e −2Φ M , and a weight-2 quasiprimary, the twisted Sugawara stress-energy tensor L tS .The latter requires non-constant Φ and hence the presence of matter.

C Dilaton-dependent measure
In this appendix, we consider yet-another alternative to the main text, where we deform the path integral measure using the dilaton field to remove the divergence in the definition of the diffeomorphism invariant path integral measure (3.3).
While this measure is not natural for minimally coupled matter fields (unless η = 0), it is diffeomorphism invariant for any η.In cases where the scalar field in 2d emerges from dimensional reduction of some higher-dimensional theory, we typically have η = 1 (though in this case, the same linear factor of the dilaton would also appear in front of the kinetic term for the scalar field in the action).
Considering our falloff conditions (2.8), the divergence problems in the definition of the measure discussed in (3. Using the dilaton-dependent diffeomorphism invariant measure (C.4), one can integrate over ψ to obtain To proceed, one would need to regularize this result using, e.g., heat kernel methods [79].
The Gauss law is violated only if there is a diffeomorphism anomaly arising at one loop.Since the classical action and the path integral measure (C.1) are both diffeomorphism invariant, the one-loop effective theory is diffeomorphism invariant.Therefore, there is no violation of the Gauss law, which implies that the Noether identities associated with diffeomorphisms are satisfied.It is still interesting to derive the Noether identities for the one-loop effective theory obtained after integrating out the scalar field ψ using the dilaton-dependent measure (C.1): L[ϕ] = L JT [g, X] + L eff [g, X] with ϕ = (X, g) and L eff [g, X] is the effective Lagrangian obtained after using the path integral measure (C.1) for the matter sector.Notably, the effective Lagrangian now depends on the dilaton field X. Repeating the steps in (5.4), we have where, as in section 5, N µ [ϕ] denotes the Noether identities and S µ ξ [ϕ] is the weakly vanishing Noether current, given by Diffeomorphism invariance implies N µ = 0 off-shell, i.e., This matches with (6.40) of the review [81].Classically, we had ∇ ν T µν = 0 on-shell for the matter sector.Semi-classically, the stress-energy tensor is not conserved due to the dilaton field.However, as opposed to the situation in section 4, this anomalous symmetry breaking does not violate (and indeed, it is even essential to satisfy) the Gauss law.

D More on Gauss law breaking
In this appendix, we provide an alternative derivation for the breaking of the gravitational Gauss law (D) using the Iyer-Wald covariant phase space methods [74].We also display more explicit formulae for this breaking.
First, we can rewrite Γ W eff [γ, χ] in Equation (4.6) to make the metric g appear explicitly: Similarly, the stress-energy tensor (4.9) in terms of the metric reads as Under diffeomorphisms, the transformation laws δ ξ γ µν = ξ σ ∂ σ (γ µν ) + γ µσ ∂ ν ξ σ + γ νσ ∂ µ ξ σ − (∂ • ξ) γ µν (D.3a) have additional terms with respect to the standard Lie derivative acting on tensors, which all involve the divergence of ξ, ∂ • ξ ≡ ∂ µ ξ µ and are only there for non-area preserving diffeomorphisms.In particular, the inhomogeneous terms in the transformation of T W µν are reminiscent of the Schwarzian derivative in a 2d CFT or similar anomalies in the sense that they depend only on derivatives of the transformation parameter ξ and on background quantities (γ and χ), but not on the transformed field.Moreover, these anomalous terms scale linearly with the central charge c.If there was no anomaly, one would expect to find a formula of the form (A.13), i.e., ω µ [ϕ; δ ξ ϕ, δϕ] = ∂ ν k µν ξ , stating that the gravitational charges are co-dimension 2. In particular, the derivative of this equation would lead to ∂ µ ω µ [ϕ; δ ξ ϕ, δϕ] = 0.However, the right-hand side (D.15) unveils the presence of an anomalous term breaking the gravitational Gauss law, confirming our result (5.12) using this alternative method.Now, as a final check, we verify that the breaking terms found in (5.12) are compatible with those obtained in (D.15).To do so, we take the derivative of (5.12) and compare it with the above formula.The breaking term in (5.12) reads as (see, e.g., [77,85]).

Figure 1 :
Figure 1: Left: Penrose diagram of AdS with reflective boundary conditions (closed system).Red lines represent propagating massless modes.Right: Penrose diagram of AdS with leaky boundary conditions (open system).External bath can be glued at AdS boundary to collect radiation.