New boundary conditions for AdS$_2$

We describe new boundary conditions for AdS$_2$ in Jackiw-Teitelboim gravity. The asymptotic symmetry group is enhanced to $\mathrm{Diff}(S^1)\ltimes C^\infty(S^1)$ whose breaking to $\mathrm{SL}(2,\mathbb{R})\times \mathrm{U}(1)$ controls the near-AdS$_2$ dynamics. The action reduces to a boundary term which is a generalization of the Schwarzian theory. It can be interpreted as the coadjoint action of the warped Virasoro group. We show that this theory is holographically dual to the complex SYK model. We compute the Euclidean path integral and derive its relation to the random matrix ensemble of Saad, Shenker and Stanford. We study the flat space version of this action, and show that the corresponding path integral also gives an ensemble average, but of a much simpler nature. We explore some applications to near-extremal black holes.


Introduction
AdS 2 plays a special role in quantum gravity because it stands as the lowest dimensional realization of the AdS/CFT correspondence [1]. It also appears universally near the horizon of near-extremal black holes [2][3][4][5][6][7][8], which suggests that AdS 2 holography will play an important role in understanding quantum black holes [9][10][11][12][13]. Despite this, the AdS 2 /CFT 1 correspondence is considered to be dynamically trivial since it was shown that AdS 2 does not support finite energy excitations [14]. More recently, after the discovery of the SYK model, it was understood that to make AdS 2 holography dynamical, one has to go to the so-called near-AdS 2 regime. This is achieved by breaking the conformal symmetry and going slightly away from the infrared fixed point [15][16][17][18], which is referred to as the nAdS 2 /nCFT 1 correspondence [19]. A canonical realization of this duality is obtained in the Jackiw-Teitelboim (JT) theory of gravity [20,21] where it was shown that the near-AdS 2 dynamics is controlled by a boundary action involving the Schwarzian derivative. The same action also governs the low-energy regime of the SYK model [22,23], demonstrating a holographic duality between JT gravity and a subsector of the SYK model.
The appearance of the Schwarzian action is tied to the pattern of spontaneous and explicit symmetry breaking Diff(S 1 ) → SL(2, R) that controls the near-AdS 2 physics. In [24], it was understood that the Schwarzian action can be seen as a Hamiltonian generating a U(1) symmetry on a coadjoint orbit of the Virasoro group. This allowed the authors to compute its partition function, which, thanks to the Duistermaat-Heckman theorem [25], is one-loop exact.
Even though JT gravity is usually thought as an effective theory, arising in the low-energy description of more complicated systems, such as higher-dimensional black holes, it can also be studied as a UV complete theory in its own right. One of the motivation to do this was to use the simplicity of this theory to probe some features of the spectral form factor, which is a diagnosis of the discreteness of the black hole spectrum [26][27][28][29]. This was considered in [30] where the full Euclidean path integral of JT gravity is computed. The authors showed that the gravitational theory is not holographically dual to a single quantum mechanical theory but rather to a statistical ensemble of theories. More precisely, this ensemble corresponds to a double-scaled matrix integral whose leading density of eigenvalues matches with the density of states of the Schwarzian theory. This leads to the more general suggestion that gravitational path integrals should be interpreted holographically in term of ensemble averages. This story has been generalized in various ways [31][32][33].
JT gravity has also played a central role in recent developments on the information paradox and the black hole interior. After introducing a coupling between an evaporating AdS 2 black hole and an external bath, the entropy of the bath was shown to follow the Page curve [34,35], using a semi-classical version of the RT/HRT/EW formula for entanglement entropy [36][37][38][39]. This led to the island prescription [40,41] which was proven using replica wormholes [42,43] appearing as saddle-points in the replicated Euclidean path integral [39,44,45]. It was noticed that these new geometries could only make sense if some kind of average was taking place.
To specify a proper classical solution space and identify the asymptotic symmetries of JT grav-ity, one needs to gauge-fix the metric. This is usually done by writing the metric in Fefferman-Graham gauge and imposing a Dirichlet boundary condition, which leads to the Schwarzian action together with its SL(2, R) symmetry. However other gauge choices and boundary conditions can be considered [46,47], leading to new boundary actions and new symmetry groups.
In this paper, we study JT gravity in Bondi gauge. The latter leads to an enhancement of the asymptotic symmetry group from the usual Diff(S 1 ) to a warped version of the latter, i.e. with an additional local U(1) symmetry. The dynamics is controlled by a generalization of the Schwarzian action, which can also be understood as the generator of a Hamiltonian symmetry on a coadjoint orbit of the warped Virasoro group. As a result, the near-AdS 2 physics is controlled by the pattern of symmetry breaking to SL(2, R)×U (1). Our boundary action matches with the low-energy effective action of the complex SYK model. This shows that our version of JT gravity is holographically dual to a subsector of the complex SYK model. We compute the full Euclidean path integral, including sums over topologies, and show that it leads to a simple refinement of the random matrix ensemble of Saad, Shenker and Stanford. We also find a connection to warped CFTs which suggests a route towards a better understanding of Kerr/CFT. Finally, we study the CGHS model as a flat space analog of JT gravity. We compute the Euclidean path integral and show that this theory is also dual to an ensemble average, albeit a much simpler one.

Summary of results
The usual boundary condition for JT gravity fixes the boundary metric in Fefferman-Graham gauge, and corresponds to a choice of AdS 2 metric of the form where t ∼ t+β and s(t) is an arbitrary periodic function. The corresponding asymptotic symmetry group is Diff(S 1 ) and acts on s(t). This infinite-dimensional symmetry gets spontaneously broken to SL(2, R) when choosing s to be The near-AdS 2 dynamics, captured by the JT dilaton, also breaks explicitly this symmetry. It is controlled by the boundary action where {f (t), t} is the Schwarzian derivative of f . This theory describes Goldstone mode fluctuations parametrized by f (t) ∈ Diff(S 1 ) around the background (1.2). In this context, the boundary action is given by the renormalized extrinsic curvature term.
In this paper, we consider a different boundary condition for AdS 2 , which corresponds to fixing the boundary metric in Bondi gauge where τ ∼ τ + β and P (τ ) and T (τ ) are arbitrary periodic functions. In Sec. 2.1, we obtain the boundary term required to make the corresponding variational problem well-defined. In this case, it is not given by the extrinsic curvature term. We show that the asymptotic symmetry group gets enhanced to the group Diff(S 1 ) C ∞ (S 1 ) which gets spontaneously broken to SL(2, R) × U(1) after we choose the thermal values where µ will be interpreted as a chemical potential for the U(1) symmetry. The explicit breaking is controlled by the boundary action This corresponds to Goldstone mode fluctuations parameterized by (f, g) ∈ Diff(S 1 ) C ∞ (S 1 ) around the background (1.5). In particular, the effective action has a U(1) symmetry, which corresponds to shifting g by a constant. The gravitational charges are computed in Sec. 2.3 and agree with the Noether charges of the boundary theory. As in the Schwarzian case, our AdS-Bondi boundary action can be reinterpreted as a particle moving on rigid AdS 2 .
In Sec. 3, we show that the Schwarzian dynamics is embedded in the Bondi description. We find that imposing an additional boundary condition that fixes P in the solution space reduces the asymptotic symmetry group to Diff(S 1 ). The boundary action (1.6) then becomes the Schwarzian action (1.3) with the relation −s(τ ) = T (τ ) + 1 2 P (τ ) 2 − P (τ ) . (1.7) This expression of s in terms of P and T is also derived by constructing a diffeomorphism that relates the solution space of Bondi gauge to the Fefferman-Graham one.
Euclidean path integral. The computation of the partition function requires an integration over a coadjoint orbit of the warped Virasoro group, which corresponds to the thermal values of P and T given in (1.5). This is greatly simplified by the fact that our boundary action generates a U(1) symmetry on this orbit. As a result of the Duistermaat-Heckman theorem, the partition function is one-loop exact, like in the Schwarzian case. The result is (1.8) Interpreting µ as a chemical potential associated to the U(1) symmetry, the partition function at fixed charge Q takes the form which follows from a Fourier transform. We see that for Q = 0, we recover the partition function of the Schwarzian theory [24]. The additional information carried by our AdS-Bondi formulation is therefore contained in this additional U(1) charge. From the partition function, we obtain the leading density of states ρ 0 (E, Q) ∝ sinh 2π 2γE − Q 2 . (1.10) Using similar techniques, we can compute the contribution of the trumpet geometry , (1.11) which also depends on the geodesic length b of the small end. This last result is a crucial step to compute the full gravitational path integral with prescribed boundary conditions. This is because any hyperbolic Riemann surface with n asymptotic boundaries can be constructed by gluing n trumpets to a genus g surface with n geodesic boundaries. This property was exploited in [30] to match the Euclidean path integral of JT gravity with a random matrix ensemble.
Our AdS-Bondi formulation allows us to fix two conditions at each boundary: the renormalized length and the U(1) charge. In Sec. 4 we show that the full gravitational path integral, including all topologies, can be computed using the following prescription: the insertion of a boundary of length β and charge Q corresponds to the insertion of Tr e −β(H+Q 2 /(2γ)) in the matrix model of Saad, Shenker and Stanford. More precisely we have where H is the random matrix while the Q i 's are scalars that shift the ground state energy at each boundary. We see that setting all the U(1) charges to zero, we recover the prescription of Saad-Shenker and Stanford.
Duality with complex SYK. The complex SYK model [48][49][50] is a version of the SYK model where the Majorana fermions are replaced by complex fermions. It is also maximally chaotic and solvable at large N , while being closer to condensed-matter systems, such as strange metals [51].
As the Majorana SYK model, it can be described at low temperature by an effective action which takes the form We show in Sec. 5 that our boundary action for JT gravity matches with the effective action of complex SYK, after a field redefinition given in (5.15). The matching between the complex SYK parameters and the gravitational parameters is given (1.14) Here, K, N, γ SYK and E are parameters of the complex SYK model while AdS , γ = κφ r and µ are gravitational parameters. The constants λ 0 and µ 0 are free parameters that appear in the identification. This matching shows that JT gravity in Bondi gauge is dual to a subsector of the complex SYK model. This duality is on the same footing as the duality between JT gravity in Fefferman-Graham gauge and the Majorana SYK model. A similar match between a flat space version of our boundary action and a particular scaling limit of the complex SYK model was achieved in [52].
Warped symmetry of AdS 2 . The AdS-Bondi gauge gives rise to an enhancement of the asymptotic symmetry group to Diff(S 1 ) C ∞ (S 1 ). In Sec. 2.3 we show that the corresponding gravitational charges belong to a centerless representation of the asymptotic symmetry algebra. However, the solution space transforms in the coadjoint representation of a centrally extended version of the group whose central charges are The corresponding algebra is the twisted warped Virasoro algebra [53]. For k = 0, the twist parameter λ can be removed by a redefinition the generators, leading to a warped Virasoro algebra with central charge c = −24λ 2 /k. In gravity, it is natural to rescale the currents P and T by κ −1 = 8πG (2) N which leads to c = 12κ, k = −2κ . (1.16) For the AdS 2 factor in the near horizon region of the extreme Kerr black hole, we have κ = J, therefore we obtain c = 12J as in Kerr/CFT [54]. Both central charges also match the ones derived for the warped symmetry of Kerr in [55]. This observation indicates that the near-AdS 2 analysis might shed light upon the lack of knowledge of the classical phase space in Kerr/CFT [56,57]. Further comments are given in Sec. 6, where we also discuss the connection to warped CFTs, following the known relation between the complex SYK model and warped CFTs [58].
Deformations of Reissner-Nordström. In Sec. 7.1, we show that the Bondi gauge for AdS 2 captures deformations of an extremal black hole in a finer way than the usual Fefferman-Graham gauge. We illustrate this for the Reissner-Nordström black hole. A deformation of the extremal geometry is generally parametrized by deviations δr + and δr − for the outer and inner horizons where r 0 is the extremal horizon. After an appropriate change of coordinate, the near-horizon geometry is obtained by taking the λ → 0 limit. In Fefferman-Graham gauge, we obtain the AdS 2 metric (1.1) in Lorentzian signature with We see that s(t) is only sensitive to the sum δr + + δr − . In AdS-Bondi gauge, we obtain the Lorentzian version of the metric (1.4) with where u is the retarded time. As a result, we see that the Bondi gauge is a finer probe of deformations of the extremal geometry. It can distinguish between the deformations δr + and δr − independently. This information is ultimately captured in the chemical potential µ associated with the new U(1) symmetry.
Embedding in near-extreme Kerr. In Sec. 7.2, we study the near-extreme Kerr black hole for which JT gravity cannot be obtained by Kaluza-Klein reduction. The linearized perturbation capturing the Schwarzian mode in near-extreme Kerr was described in [59]. We repeat this analysis and show that the Bondi near-AdS 2 dynamics described in this paper can be embedded in nearextreme Kerr. In particular, the infinite-dimensional warped symmetry algebra of Bondi AdS 2 is realized via the 4d diffeomorphism 20) which is shown to preserve a phase space of linearized perturbations described by the ansatz (7.31).
We also see that our U(1) symmetry, which was obtained in the asymptotic symmetry group of Bondi AdS 2 , corresponds here to rotations around the black hole axis in the four-dimensional geometry. The corresponding U(1) charge is then simply the change in angular momentum due to the perturbation.
Flat holography in two dimensions. A flat space version of our boundary action was derived in [52] from a modified version of the CGHS model, dubbed CGHS, which we study in Sec. 8.1. This gives rise to a flat space analog of JT gravity. 1 We show that the boundary action is equivalent to a particle moving on a rigid 2d Minkowski spacetime. We describe the thermal solution, which is a 2d analog of the Schwarzschild black hole, as depicted in Fig. 3. The corresponding symmetry breaking pattern is where the residual symmetry corresponds to a warped version of the 2d Poincaré group. We compute the corresponding gravitational charges and show that they realize a representation of the symmetry algebra. We compute the partition function which produces a linear density of states We also compute the contribution of the cylinder to the gravitational path integral. This bulk geometry is the only regular flat surface which connects two asymptotic boundaries and is depicted in Fig. 4. The result is This non-vanishing answer implies that the CGHS model should be holographically dual to an ensemble average. Let us introduce the notation Z(β 1 ) . . . Z(β n ) to represent the Euclidean path integral with n asymptotic circles of lengths β 1 , . . . , β n . The fact that the cylinder does not vanish implies that This indeed shows that the path integral should be interpreted as an ensemble average. The answer (1.23) is not the universal answer for double-scaled matrix ensembles so the dual of the CGHS model has to be something different. The only regular flat surfaces with asymptotic boundaries are the plane (or disk) and the cylinder. Therefore the path integral with an arbitrary number of boundaries is completely determined using Wick contractions involving the cylinder and the disk. This implies that the corresponding third-quantized theory is a Gaussian theory. Thus, the CGHS model constitutes an interesting example of a theory where the full Euclidean path integral can be done, while not being completely trivial and giving rise to an ensemble average. The asymptotically flat 2d black hole shown in Fig. 3 seems to be a interesting setup to study black hole evaporation and the information paradox. In contrast with the AdS setups that were studied recently, it does not require the introduction of a coupling with an external bath, because radiation can escape to null infinity. Another observation is that the simplicity of flat Riemann surfaces might constrain the existence of replica wormholes. For recent discussions on the information paradox in asymptotically flat spacetime, we refer to [60][61][62][63][64].

JT gravity in Bondi gauge
We consider near-AdS 2 gravity using Bondi gauge instead of the usual Fefferman-Graham (FG) one. Bondi gauge has been extensively studied in three and four dimensions (see [65] for a good review on both cases). We will show that in this gauge, JT gravity has an enhanced asymptotic symmetry group Diff(S 1 ) C ∞ (S 1 ) which gets broken to SL(2, R) × U(1). We will derive the boundary action and compute the gravitational charges. We will also show that this action can be interpreted as the worldline action of a boundary particle.

Boundary action for Bondi AdS
We consider the Euclidean action for Jackiw-Teitelboim gravity in two dimensions N ) −1 and the AdS radius has been rescaled to one. Since we are formulating the theory in another gauge, we need to derive the boundary term I ∂ by studying the variational problem. 2 The variation of the action at first order is given by where Θ µ is the remaining term when all the integrations by part have been made. This is the term which carries information about the boundary action. The equations of motion are Since we are considering a theory of gravity, a proper analysis of the phase space is needed in order to capture the physical degrees of freedom and to extract the symmetries that are not pure gauge. We will consider Bondi gauge, which consists in imposing two gauge-fixing conditions on the metric g rr = 0 and g ru = −1. (2.5) We start by describing it in Lorentzian signature but we will soon Wick-rotate to Euclidean signature, in which most of our study takes place. The metric takes the simple form where the r coordinate is null and u is a retarded time. The scalar curvature is R = 2∂ 2 r V . From there, we obtain the most general metric with constant negative scalar curvature in Bondi gauge where P and T are any function of the retarded time. The asymptotic Killing vectors, i.e. the vector fields which preserve the form of (2.6) on-shell, are for any function ε and η of the retarded time. The corresponding variations of P and T are The set of all the vector fields ξ (ε,η) forms an infinite-dimensional Lie algebra whose exponentiation gives the asymptotic symmetry group. The corresponding algebra is called BMS 2 and also corresponds to the symmetries of the flat version of Bondi gauge [52]. The transformations of P and T will be interpreted later in terms of a coadjoint representation of the asymptotic symmetry Lie algebra.
Having properly specified the phase space for the metric, we will use it to compute the boundary term. The metric (2.6) with V given by (2.7) is automatically solution of the equation E Φ , while the uu-component of E µν g gives Φ = ϕ 1 (u)r + ϕ 0 (u). (2.10) These two new functions also transform under the spacetime symmetry ξ The two other components of E µν g give two evolution equations for ϕ 0 and ϕ 1 From now when we say on-shell we mean that these two equations are satisfied (and their linearized version for linear perturbations). The total solution space is parametrized by four functions of the retarded time (P, T, ϕ 0 , ϕ 1 ) whose equations of motion are given by (2.12).
In the following, we would like to consider the Euclidean version of JT gravity. We will therefore perform a Wick rotation by defining (2.13) The Euclidean time lies on a circle of length β. We also make the following replacements (2.14) All the new functions are periodic in τ . These replacements are chosen so that the equations of motion (2.12) and the expressions for the field variations (2.9) and (2.11) are unchanged after the Wick rotation. From now on, we will only consider the Euclidean theory.
This being specified we deduce that the term ∂ τ Θ τ does not contribute in (2.2). For the moment we will also suppose that there is only one boundary for the AdS 2 spacetime, situated at r = ∞, so that the third term in δI JT becomes a boundary term For the variational problem to be well posed, we need B to be canceled by the variation of the boundary term in the action: when we perturb around a solution. This will ensure that solutions to the equations of motions really correspond to extrema of I JT . Consider the following boundary action One can check that, on-shell, it satisfies the following relation where the function is actually a constant on-shell. We conclude that this boundary action is practically the right one, the only thing we have to do is to impose an integrability condition. We impose so that the integrated variation vanishes and I ∂ becomes the right boundary action. The constant φ r controls the renormalized boundary value of the dilaton [16]. We define also the parameter γ = κφ r . The integrability condition is responsible for the appearance of a diffeomorphism f of the boundary circle. We impose an additional constraint which consists in fixing the zero mode of ϕ 0 /ϕ 1 whereμ is another constant held fixed in the phase space. To implement this condition, we define g satisfying g(τ + β) = g(τ ) such that The interpretation ofμ will become clear when we study the partition function. It is this condition that will give rise to the new symmetry. It will also allow for an interpretation of the solution space in terms of coadjoint orbits (a similar condition was considered in [46]). In terms of these new variables, the boundary action is The last term is a constant that can always be absorbed by a shift which maintains a well-defined variational problem. One should note also that in the term proportional toμ, only P contributes with the conditions we have on f and g. To obtain the boundary action in the form (1.6) given in the introduction, one should perform a redefinition of the fields described in Sec. 2.4.
We have imposed additional conditions on the solution space, we therefore need to check if they are affecting the asymptotic symmetry group. The first condition does not lead to a change of the symmetry algebra but the second one requires that η = σ , with σ satisfying the same condition as g. The new transformations are δ ξ f = εf , δ ξ g = σ + εg .
(2. 25) We see that ε is an infinitesimal reparametrization, while σ acts by shifting g. We will now study this symmetry algebra in more details. This will be the occasion to review some mathematical results which are useful to understand the solution space and the boundary action.

From BMS 2 to warped Virasoro
As we said earlier, the set of all the vectors ξ (ε,η) forms an infinite-dimensional algebra called BMS 2 and the associated bracket is This algebra corresponds the finite spacetime coordinate transformations where F is a reparametrization of the circle while H is an arbitrary periodic function. They are very similar to the BMS 3 transformations at null infinity in three dimensions, see [65]. A major difference is that the Euclidean time here plays the role of the celestial angle there. This has important consequences on the interpretation of the boundary theory. We will use the same terminology as the one used for BMS 3 to describe the transformations, F will be called a boost while H will be called a supertranslation.
The conditions that we have imposed on the phase space give a constraint on the supertranslation, so that the finite transformations become where G are arbitrary periodic functions. The corresponding algebra is spanned by the vectors which satisfy the algebra If we define the modes L n = iβ 2π e 2πint/β , 0 and J n = 0, β 2π e 2πint/β , it becomes which is known as the warped Witt algebra [53]. This algebra can be centrally extended The corresponding group is the semidirect product of the diffeomorphisms of the circle with the smooth functions on the circle: Diff(S 1 ) C ∞ (S 1 ). It has the same structure as BMS 2 but the group law induced by the coordinate changes (2.28) is different The coadjoint representation of this group will play an important role in what follows, and this is why we want to study it here. We will describe only the minimum in order to be self-contained, for a complete mathematical description see [66]. An element of the Lie algebra will be denoted v and we have v = (τ )∂ τ + σ(τ ) + a i e i . (2.34) The constants a i 's account for possible central extensions. The dual algebra, which is the space of forms on the Lie algebra is then spanned by covectors where P and T are any function on the circle. The c i 's are constants and the e i 's satisfy e i (e j ) = δ i j . The action of b on v is then given by the bracket (2.36) Having defined the action of covectors on vectors, we can define the action of a group element on the covector b, which is the coadjoint representation The action of a group element on b induces a transformation of the two functions P and T that are interpreted as currents. Exactly like the holomorphic and anti-holomorphic components of the energy-momentum tensor in a 2d CFT which transform in the coadjoint representation of the Virasoro algebra. The coadjoint representation of the warped Virasoro group is described in [67] and corresponds to the transformations (2.38) The three constants c, k and λ are all the possible central extensions of the warped Virasoro group. We have defined the Schwarzian derivative (2.39) The functions T and P being periodic, we can define the generators L n and J n to be the modes of T and P on the circle. The centrally extended algebra (2.33) is then recovered for the bracket These transformations for P and T follow directly from group theory. We can also view the group as acting on the spacetime coordinates like in (2.28). This action on the Euclidean Bondi metric also induces finite transformations of the functions P and T appearing in the uu-component. Acting on the bulk metric with (2.28), we find that they correspond exactly to the coadjoint transormation (2.38) for the central charges c = 0, k = −2 and λ = −1. (2.40) The transformations (2.9), with η = σ , are the infinitesimal version of these transformations. The fact that the solution space transforms in the coadjoint representation of the asymptotic symmetry group is mysterious but not rare. It also happens for the usual boundary conditions for AdS 2 in Fefferman-Graham gauge as we will see later. This was also shown for a version of the CGHS model in [52]. Another important example is the case of 3d gravity with the Brown-Henneaux boundary conditions, for which the solution space is parametrized by the energy-momentum tensor of a 2d CFT. But there is even more to say about the relation between the coadjoint representation of the warped Virasoro group and 2d gravity in AdS-Bondi gauge. We have found that the bulk action reduces to the boundary term when evaluated on the Bondi AdS solution space, and when the integrability conditions are taken into account. Interestingly we recognize also the coadjoint action of a group element (f, g) on T in the first integrand and on P in the second one, such that the boundary action becomes where we have definedτ = f (τ ). One should bear in mind that this time, the group element does not come from a coordinate change, but was really defined by the dilaton through the integrability conditions. The solution space of Bondi-AdS is therefore parametrized by a group element (f, g) of the warped Virasoro group and a vector in the dual Lie algebra defining two currents P and T . The action is simply given by the coadjoint action of the group element on T and P . The transformations (2.25) are nothing but an infinitesimal version of the group law. Indeed, taking gives the linearized version of the group law We will make use of this property later when studying the symmetry breaking.

Gravitational charges
In gauge theories, charges associated with asymptotic symmetries can be constructed using the covariant phase space formalism [68]. This gives a way to define surface charges associated to diffeomorphisms that are not pure gauge because of the presence of a boundary. The diffeomorphisms that we study here are given in (2.28).
At first one constructs the field variation of a charge (which corresponds to a one-form in the field configuration space) in the following way. Starting with the symplectic potential the field-exterior derivative defines the symplectic form where we have definedφ 0 = iφ r g /f . The sympetic form ω ω ω is a 2-form in the field space. Now the fundamental theorem of the covariant phase space formalism tells us that when φ and δφ are on-shell, there exists a function k ξ such that where d is the spacetime exterior derivative. The function k ξ is a one-form in the field space. It can always be decomposed in the following way and we say that k ξ is integrable when Ξ ξ vanishes. Usually, when we have integrability, k ξ is integrated over a codimension-two surface in order to define a charge. In two dimensions, this would just be a point so we leave it as it is. For we find that k ξ depends only on τ and is given by Which can be decomposed into an integrable and non-integrable part in the following way (2.52) The usual representation theorem states that when the charges are integrable, the bracket defines a representation (possibly centrally extended) of the asymptotic symmetry group. Here, δ χ Q ξ means that we take the field variation of the charge Q ξ , and replace δφ by δ χ φ. When the charges are not integrable, one can define the modified bracket This bracket was introduced in [69] where it was used to define a centrally extended representation of the BMS algebra in 4d asymptotically flat gravity. It was also used in the context of near-horizon symmetries in [70,71]. One can show that in our case it also defines a representation where the bracket on the right hand side coincides with (2.30). The central extension is actually trivial since it can be absorbed in the following redefinition of the charge which becomes in terms of f and g, From the representation theorem, we can deduce the time evolution of the charge. Using the vector (ε = 1, σ = 0), the associated variations are all time derivatives δ (1,0) φ = φ . We replace one of the two vectors by (1, 0) in (2.55) to obtain The non conservation of the charge is sourced by the non integrable part. We conclude that if we find a proper restriction of the phase space on which Ξ ξ vanishes, the associated symmetries will be true symmetries.
We would like to implement such a restriction on the solution space. An obvious choice is to require that the two currents are constant P (τ ) = P 0 and T (τ ) = T 0 , (2.59) held fixed in the solution space. The infinite-dimensional symmetry algebra is then broken to the subalgebra that leaves P 0 and T 0 invariant. Eq. (2.9) becomes Solving for ε and σ, we find where we have defined −s 0 = T 0 + 1 2 P 2 0 . For the transformations associated to λ 2 and λ 3 to be well-defined (not considering winding), we need to have Then, the transformations associated to λ 1 , λ 2 and λ 3 generate an SL(2, R) symmetry. Moreover there is an extra U(1) symmetry corresponding to shifting of g by a constant, which is λ 4 here. Therefore, requiring P and T to be constant and to satisfy the relation (2.62) realizes the symmetry breaking We have found that using this boundary condition, which leads to integrable charges, the asymptotic symmetry group is bigger than the vacuum symmetry group SL(2, R). 3 Morevover, the gravitational charge associated to the U(1) symmetry is This additional symmetry will play an important role in the study of the Euclidean path integral. Moreover, one can show that with this condition, the gravitational charges agree with the Noether charges of the boundary action (ignoring the term proportional toμ which does not contribute to the dynamics). One should note that the global U(1) transformation (ε = 0, σ = 1) has actually no real effect on the initial fields as it leaves Φ and g µν invariant. This however does not mean that it is not a true symmetry. In fact, such symmetries arise frequently and are known as reducibility parameters. They act trivially on the fields on-shell but have a non trivial charge (see the generalized Noether's theorem in [72]). 4 In what follows we will also consider geometries for which s 0 = 2π 2 β 2 (in particular the trumpet geometry). In that case, the transformations associated to λ 2 and λ 3 are not defined so the symmetry breaking pattern is where the unbroken symmetries correspond to the zero modes of ε and σ.

Particle interpretation
We consider again the boundary action (2.41) and make the change of variableτ = f (τ ) together with the field redefinitionf = f −1 andg = −g • f −1 . The boundary action becomes with the following Lagrangian This can be interpreted as describing the dynamics of fluctuations (f, g) ∈ Diff(S 1 ) C ∞ (S 1 ) around the background geometry As for the usual Schwarzian action, this boundary action can also be understood as the motion of a boundary particle [16,74]. In this picture, the full dynamics is captured by the motion of a particle near the boundary of rigid AdS 2 . The simplest way to obtain this is to consider a "vacuum" boundary particle whose trajectory is at a constant r = r 0 . Applying the diffeomorphism we obtain a new trajectory depending on (f, g). The worldline action of the corresponding boundary particle on the geometry (2.68) takes the form We consider a particle whose trajectory is the image of the vacuum one and compute the corresponding Lagrangian, we obtain Up to a constant term, the particle Lagrangian is the same as the one appearing in the gravitational boundary action (2.67). Intriguingly, the match is achieved with the identificationμ = r 0 . When studying the Euclidean path integral in Sec. 4, we will see thatμ is interpreted as the chemical potential associated with the U(1) charge. The identification with the boundary particle suggests that this U(1) charge is related to the radial direction of AdS 2 . This particle usually lies at r 0 → +∞, i.e. close to the asymptotic boundary of AdS 2 . It is interesting to note that we do not need to take such a limit here to match the boundary action with the particle action. This suggests that there might be a relation with the finite cutoff versions of JT gravity discussed recently [75,76].

Relation to the Schwarzian
In this section, we show that the boundary action derived in the previous section can be seen as a generalization of the usual Schwarzian action. We start by writing a diffeomorphism that maps the solution space of Bondi gauge to the one FG gauge with the usual Dirichlet condition. With an additional boundary condition in Bondi AdS, we recover the Schwarzian action starting from the one derived in Bondi gauge (2.41). We also show how the asymptotic symmetries reduce to one copy of the Virasoro group and how to recover the FG gravitational charges from the Bondi ones given in (2.52). The computation of the partition function in Sec. 4 will show that the Schwarzian theory is actually a subsector of the Bondi theory, corresponding to setting the U(1) charge to zero.

From Bondi to Fefferman-Graham
We start by writing the bulk metric in FG gauge. The metric takes the simple form The most general solution to the equation R = −2 is The usual boundary condition is h(t) = 1. It is the equivalent of the Brown-Henneaux boundary condition in two dimensions, i.e. requiring the boundary metric to be flat. Looser boundary conditions were considered in [46], where h was allowed to fluctuate. When asking h(t) = 1, the asymptotic Killing vectors are where ε is any function of the time direction. The asymptotic symmetry group is therefore isomorphic to Diff(S 1 ). The effect of the Killing vector on the metric is to modify the function s in the following way δ ξ s = εs + 2sε + ε .
where we have c = 12.
An analysis similar to the one realized in Sec. 2.1 was done in [46] for the Schwarzian action. They derive the boundary term by demanding a well-defined variational problem. We are going to take another route here, which consists in recovering the Schwarzian action from the boundary action in Bondi gauge.
We would like to find a diffeomorphism that maps the metric in Bondi gauge (2.68) to the metric in FG gauge (with h(t) = 1). We start by going in tortoise coordinates followed by We can solve order by order in r −1 to find the coefficients 5 . The first ones are The resulting metric is indeed in FG gauge with a function s written in terms of P and T : Compared to the Schwarzian action, the boundary action in Bondi gauge has an additional mode g and a new physical symmetry. To recover the Schwarzian action, we need to kill this additional mode by imposing a further boundary condition in Bondi gauge. We choose to impose P (t) = P (t) where P is fixed in the solution space. This translates into a condition on the asymptotic Killing vectors. The first equation of (2.9) gives σ = ε P + εP + ε , (3.10) so that the Bondi asymptotic Killing (2.29) is now parametrized by ε only. We can deduce the transformation of s from the second equation of (2.9) and (3.9) This is exactly the transformation we have obtained from the asymptotic Killing vectors in FG gauge. Now, we would like to impose this further boundary condition at the level of the boundary action and write it in terms of the function s. We recall that we have two remaining equations of motion (2.12) for the dilaton components ϕ 0 and ϕ 1 . In terms of the group element (f, g) they are (3.12) 5 The same notation has been used for the radial coordinate in Bondi and FG gauge.
Solving the first equation for g and using the definition of s in (3.9) (together with the two conditions (2.20) and (2.22) on the solution space), we can rewrite the boundary term (2.17) in terms of f and s as The integrand is exactly the coadjoint action (3.5) of f on the current s. We observe the same structure as the one discussed in the previous section. The only difference is that the group has changed. Asking the current P to be held fixed in the solution space removes the shift symmetry so that the symmetry group reduces to Diff(S 1 ). Interestingly, the integrand of the boundary action (after integrating one of the equations of motion) remains written in terms of the coadjoint action of the central extension of Diff(S 1 ), i.e. the Virasoro group.

Gravitational charges and the Schwarzian
Having found the mapping from Bondi to FG, we are also able compute the gravitational charges.
To do so, we consider the charges found in Bondi gauge (2.52), and impose the condition P = P with the corresponding condition on the symmetries (3.10). After integrating out the first equation of (3.12), we obtain (up to a constant) (3.14) Again, one can show that under the modified bracket (2.54), these non-integrable charges belong to a representation the Diff(S 1 ) algebra. A consistent condition to have integrability is to impose s(τ ) = s 0 , a constant held fixed in the solution space. This condition must be preserved by the asymptotic symmetries. From (3.11) we obtain We recover the SL(2, R) symmetry of AdS 2 for s 0 = 2π 2 /β 2 , but we do not have the extra U(1) symmetry anymore. With the above choice for s 0 , we define f −1 =f and make the change of variableτ = f (τ ) to obtain the action This is the usual Euclidean Schwarzian action. The finite version of the symmetry (3.16) corresponds to the SL(2, R) symmetry of the Schwarzian derivative where F = tan π βf . Thus, requiring s to be constant realizes the symmetry breaking from Diff(S 1 ) to SL(2, R). Moreover, one can show that the charges (3.14) give three charges, one for each generator of (3.16), which precisely match with the Noether charges of the Schwarzian action.

Euclidean path integral
We would like to compute the Euclidean path integral of the boundary action of Bondi AdS. We start with the action given in (2.67) which we reproduce below In the following we will take constant values P = P 0 and T = T 0 . The on-shell action is obtained for f (τ ) = τ and g(τ ) = 0 and reads where we have defined µ ≡ βμ which will be interpreted as the chemical potential associated to the U(1) symmetry. The path integral is defined as 3) The fields f and g belong to the warped Virasoro group and the action has a symmetry that corresponds to the stabilizer of P 0 and T 0 under the coadjoint action of the group. Therefore we should integrate over the manifold This manifold is generically infinite-dimensional and is isomorphic to the coadjoint orbit of the covector (P 0 , T 0 ). It can be endowed with a canonical symplectic form which provide a measure for the path integral. We will start by computing the partition function on geometries that correspond to the hyperbolic disk. These geometries need to satisfy we recall that s 0 is the variable appearing in the geometry in FG gauge. The value of P 0 is fixed onshell in terms of the constantμ which appeared in the boundary action. This follows from dividing the first equation in (2.12) by ϕ 1 and integrating over the circle, then using the boundary condition (2.21) it leads to The relation (4.5) then fixes T 0 and we get for the disk We see that we have a one-parameter family of configurations, labeled by µ. Note that a particular configuration can be generated by acting with the diffeomorphism (2.28) with on the Euclidean metric with P 0 = 0 and T 0 = 2π 2 /β 2 . The function G is not periodic, therefore the couple (F, G) does not belong to the group. This means that µ parametrizes a family of inequivalent coadjoint orbits. Each of them define good phase spaces but they are still physically inequivalent according to the study of gravitational charges in Sec. 2.3. The disk partition function is then computed as a path integral over f, g for the action defined with the above values of P 0 and T 0 . As a result, it depends on β and µ Z(β, µ) = Df Dg e −I . (4.9) The parameter β appears because we are fixing the circle to have renormalized length β. The parameter µ appears as an arbitrary parameter labeling a family of possible boundary actions. We will interpret below the parameter µ as the chemical potential associated to the U(1) symmetry.

Cardy thermodynamics
Before delving into the computation of the exact partition function. It is instructive to consider the thermodynamics in the saddle-point approximation. This derivation will be similar to the derivation of the Cardy formula in 2d CFT. In fact, we point out in Sec. 6 that there is a precise match between the entropy of JT in Bondi gauge and the entropy of warped CFTs.
We assume that the free energy is approximated by the on-shell action which, as will be shown later, is achieved when β is small. For a thermal state, the on-shell action is computed using the values for P and T given in Eq. (4.7) and the classical solution corresponds to f (τ ) = τ and g(τ ) = 0. This gives the free energy The entropy is obtained using a Legendre transform The saddle-point value of the energy and the charge are given by Inverting these relations gives the temperature and chemical potential at the saddle-point (4.14) This allows to rewrite the entropy (4.12) as The saddle-point approximation is valid for large energy and charge. After computing the full partition function, we will see that we reproduce this entropy by taking the Cardy limit of the exact density of states (4.48). In Sec. 6, we match this entropy with the Cardy formula of a warped CFT. In Sec. 5, we also match it with the entropy of the complex SYK model.

Partition function
We now consider the computation of the exact partition function. 6 The Kirillov-Kostant-Souriau symplectic form associated to coadjoint orbit of the warped Virasoro group was obtained in [77], it is given by where we have definedg = g − i µ β f − log f . Here, α is a normalization constant for the symplectic form. As in [24], we can use the Duistermaat-Heckman theorem that states that if the action generates a U(1) symmetry on the integration manifold then the partition function is one-loop exact. Indeed one can check that we have which means that I generates a time translation symmetry on the coadjoint orbit. 7 To organize the path integral, it is useful to write it as a classical contribution and a one-loop contribution The classical contribution is given in (4.2) and gives with the values (4.7) The one-loop part is obtained by integrating over the fluctuations ε(τ ) and σ(τ ). The quadratic action is given by (4.20) We will start with generic values of P 0 and T 0 since some of these results will be useful when computing the partition function on the trumpet. The boundary conditions impose that ε and σ are periodic with period β. Hence, they can be decomposed into modes Since ε(τ ) is real and σ(τ ) is pure imaginary, we have the relations Using the decomposition, we can write the one-loop integral as We decompose the symplectic form using the modes We note that the zero mode ε 0 and σ 0 do not appear and are hence degenerate directions of the symplectic form. This corresponds to the U(1) × U(1) symmetry of the coadjoint orbit. As they are degenerate directions, they must be removed in the computation of the partition function. We decompose ω into blocks The Pfaffian takes the form Making use of the identity we obtain We also compute which gives Putting everything together gives We can write the Pfaffian as Pf n (ω), Pf n (ω) = α 2 β 2 n 2 n 2 + β 2 2π 2 We also decompose the quadratic action (4.20) into modes I quad = γβ n∈Z T 0 n 2 ε n ε −n − 1 2 n 2 σ n σ −n + P 0 n 2 ε n σ −n − 2πi β n 3 ε n σ −n . (4.34) We restrict the sums to n ≥ 1 using the constraint (4.22) and decompose the modes into their real part and imaginary part This allows us to write where We can perform the Gaussian integral and we obtain 8 The analysis above was done for arbitrary values of P 0 and T 0 and will be useful later, when we consider the trumpet. We now evaluate the Pfaffian for the values of P 0 and T 0 corresponding to the disk. From the relation we obtain where we have removed the m = 1 contribution which would be degenerate. This is because the coadjoint orbit has an enhanced symmetry SL(2, R) × U(1), corresponding to the modes ε ±1 . The Pfaffian for the disk can be written as (4.42) 8 Here, we have The one-loop path integral can be decomposed as where we compute using zeta regularization for the infinite product Finally combining everything, we obtain the disk partition function Interpreting µ as a chemical potential associated to our new U(1) symmetry, we have a density of states ρ 0 (E, Q) defined from the formula This density of states is the same as the Schwarzian density of states but with a shifted ground state energy. We see that the Schwarzian theory corresponds to the Q = 0 sector of the Bondi theory. Another way to see it is to consider the mixed ensemble partition function obtained using an inverse Laplace transform We see that for Q = 0 we recover the Schwarzian partition function. For Q = 0, the Fourier transform is just an integration over µ, which can be interpreted gravitationally as a sum over inequivalent fluctuations of the same geometry: the hyperbolic disk. The dynamics of these fluctuations is governed by an action whose coupling constants depend on µ. From a more mathematical perspective, the integration over µ can also be seen as a sum over non-equivalent coadjoint orbits.

Genus expansion and matrix integrals
In the previous section, we computed the Euclidean partition function of the boundary theory. This can also be seen as the gravitational path integral on the disk geometry. In this section, we consider the full gravitational path integral, including the sum over all possible Euclidean geometries with fixed boundary conditions. For JT gravity in Fefferman-Graham gauge, it was shown in [30] that the gravitational path integral was given by correlations functions of a double scaled matrix model. We will see that our new boundary conditions leads to a simple generalization of this result.

Gravitational genus expansion
The full Euclidean JT action We have included the pure gravity term where Φ 0 is a constant which sets the extremal entropy S 0 = 2πκΦ 0 . 9 This term is topological in two dimensions. The action I JT contains a boundary term for the variational problem to be well posed. It ensures that the solutions to the equations of motion are indeed saddle point of the action. When considering the Bondi phase space it will simply reduce to one copy of our boundary action for each circle.
Consider the gravitational path integral on geometries with multiple boundaries. On each boundary i = 1, ..., n we fix the length β i and the chemical potential µ i . The gravitational path integral becomes schematically since we have I top = −S 0 χ = −S 0 (2 − 2g − n). Now each Z g,n corresponds to the path integral on a fixed topology. The integration over the dilaton is crucial since it imposes the constraint R = −2.
The bulk term of I JT vanishes in the exponential so that path integral becomes (4.53) The first integral corresponds to the sum over inequivalent hyperbolic Riemann surfaces with g handles and n boundaries. The second integration corresponds to a sum over the fluctuations associated to the boundary action, where of each boundary the values of β and µ are fixed. We can in the matrix ensemble.
equivalently fix the the charge Q i at each boundary using Fourier transform (4.54) We will need to consider two types of boundary fluctuations: on the disk and on the trumpet. This is because any hyperbolic Riemann surface with n asymptotic boundaries can be constructed by gluing n trumpets to a genus g surface with n geodesics boundaries. A typical example of such a geometry is depicted in Fig. 1. The case g = 0 is special since there is only the disk. The disk has no modulus, while for the trumpet there is the length b of the geodesic at the small end, see Fig. 2. We obtain (4.55) The constant α multiplies the symplectic form 10 . It will ultimately be absorbed in a redefinition of S 0 . The function V α g,n (b 1 , . . . , b n ) is the volume of the moduli space of hyperbolic Riemann surfaces with prescribed geodesic boundaries. The only new feature compared to the computation in [30] is that our definitions for the disk and the trumpet contributions are different since we have this additional boundary chemical potential µ. The geometries do not change, they are characterized by the value of the current s 0 when we write them in FG gauge. For the disk it is s 0 = 2π 2 β 2 , while for (β, Q) b Figure 2: The trumpet geometry: we specify the boundary conditions β and Q at the end of the asymptotic boundary. The small end is labeled by the length b of the geodesic boundary.
the trumpet, which is some sort of hyperbolic disk with a conical defect, it is s 0 = − b 2 2β 2 . To obtain these values, one can compute the on-shell value of the Schwarzian action on these geometries.

Trumpet geometry
We would like now to compute the trumpet contribution Z trumpet . From the relation and the relation (4.6) between P 0 and µ, we obtain the values The on-shell action (4.2) evaluates to In this case, the Pfaffian (4.32) takes the form Pf n (ω) . The computation is very similar to the disk, which was considered in the previous section. The difference is that we must not remove the two modes ε ±1 because they are not degenerate directions of the symplectic form anymore. The integration is now over the infinite-dimensional manifold where two U(1)'s correspond to the zero modes ε 0 and σ 0 . To compute the one-loop contribution, we can use the formula (4.39), which is valid for a general values of P 0 and T 0 . We obtain Pf n (ω) d 2 ε n d 2 σ n e −I (n) quad = 2πγ βα , where we have used the formula (4.45) for the infinite product. Combining the one-loop contribution and the on-shell action, we obtain the trumpet contribution (4.62)

Matrix integrals
The genus zero contribution Z disk was computed in the previous section, see Eq. (4.49). To have the same normalization as [30], we make the following redefinition of the disk partition function at fixed Q This just corresponds to a redefinition of the chemical potential and the charge We obtain We see that for Q = 0, this coincides with the partition function of the Schwarzian action on the disk. We redefine in same way the trumpet contribution to obtain For Q = 0, this matches with the Schwarzian partition function on the trumpet [30]. We now have all the ingredients to give the final result. Before, we should make a comment this normalization constant α. The overall α dependence of Z g,n is α n · α 3g+n−3 · α −n/2 = α −3χ/2 . (4.67) Since the power of α is proportional to the Euler characteristic χ, we can absorb α in a redefinition of the extremal entropy S 0 and set α = 1.
We denote with the superscript SSS the quantities of Saad-Shenker-Stanford [30]. Here, they correspond to setting to zero all the U(1) charges. Let's consider the example of the double trumpet, obtained by gluing two trumpets along their geodesic boundaries (or small ends). Its contribution is computed by integrating over the modulus b. 11 We find 11 The measure is bdb because we also have the freedom to twist one trumpet with respect to the other.
where Z SSS 0,2 (β 1 , β 2 ) = √ β 1 β 2 2π(β 1 +β 2 ) . The difference with [30] is that each boundary receives a contribution from the corresponding U(1) charge. This contribution corresponds to a multiplicative factor at each boundary. Indeed, for a generic number of boundaries, we have where the average is taken in the same matrix ensemble. Note that the effect of the Q i 's is to shift the energy at each boundary. We emphasize that the charges Q i are not matrices but scalars here.
A similar structure appears when boundary global symmetries are added to the matrix ensemble of JT gravity [33,78]. This can be realized by adding a BF theory in the bulk. In this setting, each boundary is labeled by an irreducible representation r i of the corresponding group. The result is that there is also a factorization The similarity is that the gauge symmetry in the bulk also shifts the ground state energy, by the Casimir c 2 (r) of the representation, which is indeed proportional to Q 2 for us. The difference is that in our model, we can have different charges at different boundaries of a connected geometry: there is no constraint enforcing equality of the charges. This stems from the fact that in the trumpet, there is no charge at the small end of the trumpet, which was possible in [33,78] due to the presence of bulk gauge fields. Our model only contains pure gravity and all the dynamics reside at the boundary.

Duality with complex SYK
In this section, we show that JT gravity in Bondi gauge is dual to the complex SYK model. This is done by giving the precise match between the effective actions. This duality says that the complex SYK model contains a subsector describing JT gravity in Bondi gauge, in the same way that the Majorana SYK model contains JT gravity in FG gauge as a subsector. A similar match between a flat space version of our boundary action and the complex SYK model was given in [52].

The complex SYK model
The complex SYK model [48][49][50] is a generalization of the SYK model [19,79], obtained by replacing the Majorana fermions with complex fermions. This model is of special interest because it is maximally chaotic and solvable at large N , but closer to condensed-matter systems than the original SYK model. For example, it was recently used to investigate strange metals [51].

The complex SYK model is a quantum mechanical model involving a large number N of complex fermions with a random interaction. The Hamiltonian is
where A{. . . } denotes the antisymmetrized product of operators. The couplings J i 1 ...iq are independent random complex variables with zero mean and variance where q is an even integer greater than four. There is a global U (1) charge We also define the specific charge Q = 1 N Q which is related to the UV asymmetry of the Green function In the IR, the spectral asymmetry parameter E characterizes the long-time behavior of the zero temperature Green function Here, S(Q) is a universal function, in the sense that it is fully determined by the structure of the low-energy theory. At small but non-zero temperature T = β −1 , the complex SYK model is described by an effective low-energy action given by [48] where f (τ ) is the Schwarzian mode that reparametrizes time and g(τ ) is a U(1) mode which is periodic in the absence of winding. This action is the analog of the Schwarzian action for the Majorana SYK model. In comparison, the complex SYK model has an additional U(1) mode. The parameters γ SYK and K are defined from thermodynamical properties: γ SYK characterizes the linear response in temperature of the entropy While K is the zero temperature compressibility defined as where µ SYK is the chemical potential associated with the U(1) charge. The spectral asymmetry parameter is also related to the zero temperature entropy according to dS(Q) dQ = 2πE . (5.10)

Holographic duality
Symmetries. A connection between the complex SYK model and warped CFTs was explained in [58]. It was observed that the complex SYK model has an underlying warped Virasoro symmetry which is spontaneously and explicitly broken down to its global part This is the direct analog of the spontaneous and explicit breaking of Diff(S 1 ) → SL(2, R) in the Majorana SYK model. The symmetry breaking pattern (5.11) of the complex SYK model is the same as the symmetry breaking which controls the version of JT gravity described in this paper. This is the first hint of a relation with our Bondi-AdS version of JT gravity and the complex SYK model.

Action.
In the rest of this section, we will show that the effective action precisely matches with our boundary action. A similar matching between the action of complex SYK and the CGHS model was achieved in [52] in the context of flat holography. This matching was involving a special scaling limit, which should correspond to the flat limit of our theory. Note that a different holographic interpretation of complex SYK, with a 2d gauge field, was proposed in [80].
We start with our boundary action for AdS-Bondi JT gravity where we have restored the AdS 2 radius AdS . We recall that γ is written in terms of the renormalized value of the dilaton and the Newton constant as follows We consider this theory at a temperature β and chemical potential µ. This corresponds to the choice where we have reinstated the factors of the AdS 2 radius AdS . We now redefine g according to where λ 0 and µ 0 are two arbitrary parameters. Note that this redefinition changes the periodicity of g since we haveg We can take µ 0 = 0 to obtain a functiong that remains periodic. The action can then be written as This is precisely the action of the complex SYK model (5.7) with parameters We can also report the gravitational parameters in terms of the SYK parameter We see that N controls the gravitational coupling and the AdS 2 radius. We note that the value µ 0 = µ corresponds to E = 0 (this value for µ 0 is special because it diagonalizes the symplectic form, see (3.9) of [77]). We have also reported the value of λ 0 , which is the other arbitrary parameter in the matching. We have shown that the AdS-Bondi JT gravity described in this paper is dual to the complex SYK model. This holographic duality is on the same footing as the duality between the standard JT gravity and the Majorana SYK model. That is, the gravitational description corresponds to a low-energy subsector of the complex SYK model.

Thermodynamics.
It is instructive to also match the large N entropy of the SYK model. Following [58], the entropy of the complex SYK model can be written where we have defined Q SYK = Q . In comparison, the entropy of JT gravity computed in (4.48) leads at large 2 2 AdS γE − Q 2 to With the choice of parameters (5.18), we get a precise match provided the relation The parameter λ 0 , which appeared as an arbitrary parameter in the matching of the actions, has the interpretation of a relative rescaling of the U(1) charges between gravity and the SYK model. It is also interesting to see that the leading logarithmic correction to the entropy can also be matched. In the grand canonical ensemble, the expression (4.46) for the partition function shows that we have the correction δS JT = −2 log β . (5.23) This matches with the logarithmic correction to the corresponding entropy of the complex SYK model, as reported in [50]. We should make a comment on this matching of the entropy. In Sec. 4.2, the partition function (4.46) of JT gravity in Bondi gauge is not exactly the same as the partition function of the complex SYK model considered in [58]. This is because our boundary action (2.67) has an additional term, involving the coadjoint transformation of P , which does not contribute to the dynamics but gives an additional constant in the partition function. The effect of this constant is to change the sign of the µ 2 term in the exponential, and hence corresponds to the replacement µ → iµ. This change can be absorbed in a redefinition of the relation between the partition function and the density of states, leading to the same formula for the density of states and therefore the same entropy.

Relation to warped CFTs
The near-horizon geometry of any extremal black hole has an SL(2, R) × U(1) symmetry. This was one of the motivation to introduce the notion of warped CFTs in [53]. warped CFTs are theories which are analogous to 2d CFTs but whose symmetry algebra is the warped Virasoro algebra, which consists in one copy of the Virasoro algebra together with a U(1) Kac-Moody algebra. The corresponding global symmetry is SL(2, R) × U(1). In this section we describe a relation between AdS 2 gravity in Bondi gauge and warped CFTs (see [81] for an early discussion on the subject). We will also comment on some intriguing observations related to the Kerr/CFT correspondence.
For a warped CFT whose coordinates are t and φ, the warped conformal symmetry corresponds to the following change of coordinates In our setup, the coordinate φ becomes the Euclidean boundary time of the AdS 2 spacetime while the coordinate t has no immediate gravitational interpretation. In this context, the quantities T (φ) and P (φ) are the associated conserved currents; they transform like in Eq. (2.38) with the replacement τ → φ. The modes of T and P , denoted respectively L n and J n , satisfy the algebra (2.33) for the bracket . The central charge λ, called the twist parameter, can always be absorbed in a redefinition of the generator L n [52] L n → L n + 2iλ k nJ n , We would like to compute the vacuum values of P and T on the cylinder. The mapping between the plane and the cylinder is realized by the coordinate change [53] t → t − α β φ, φ → e 2πiφ/β , (6.4) which leads to the thermal identification Setting the vacuum values on the plane to zero and using the transformations (2.38), we obtain the vacuum values on the cylinder We define the effective chemical potential α * = α − 2λ k (2πi/β). This redefinition of the chemical potential allows to rewrite the vacuum values of P and T in terms of the new central charges c * , λ * and k * as We see that the twist parameter has been absorbed in the redefinitions of the central charges and the chemical potential. With our values for the new central charges (6.3) and the identification α * → µ the vacuum values become They correspond exactly to the values of T and P we have been using to compute the gravitational path integral at fixed temperature and chemical potential in Sec. 4. In the AdS 2 bulk, the map from the plane to the cylinder is realized by the following diffeomorphism The connection between our boundary action for AdS 2 and warped CFTs follows from the results of [58]. There, it was shown that the partition function of the complex SYK model matches with a particular limit of the vacuum character of a warped CFT. This is similar to the relation between the Schwarzian partition function and the vacuum character of a CFT 2 [82]. This also leads to a matching of the leading thermodynamics as explained in [58]. We find that the thermodynamics of JT gravity in Bondi gauge matches with that of a warped CFT in the Cardy regime. The entropy of a warped CFT can be written as where in the Cardy regime, we have This was derived in [53] and here, we follow the conventions of [58]. Using our values for the central charges (6.3), and writing the quantum numbers as where γ = κφ r is the parameter controlling the deviation from extremality, we obtain The right moving entropy S R matches with the entropy of AdS-Bondi JT gravity given in (4.15). As in [58], we interpret S L as contributing to the ground state entropy. It can also be noted that the one-loop correction to the entropy in a warped CFT gives also a logarithmic correction δS = −2 log β , (6.14) which agrees with our Euclidean path integral computation.
We would like to make a final comment on the central charges in Eq. (6.3) and a potential application to the Kerr/CFT correspondence. In the gravitational context, it is natural to rescale P and T by κ −1 = 8πG (2) N to obtain physical central charges multiplied by κ For the AdS 2 spacetime appearing in the extreme Kerr black hole, we have κ = J 12 so our central charge reproduces the central charge c = 12J [54]. A warped Virasoro symmetry has also been described for Kerr in [55] and we reproduce the central charges c = 12J and k = −2J that were obtained there. A major shortcoming of the Kerr/CFT approach is a lack of knowledge of the classical phase space on which the symmetry algebra is acting. We hope that the near-AdS 2 realization of these symmetries will shed light on this issue.

Near-extremal black holes
Extremal black holes have a universal AdS 2 factor in their near-horizon geometries. This makes near-AdS 2 holography a nice framework to understand near-extremal black hole dynamics. In this section, we will demonstrate the relevance of our boundary conditions in this context. We will show that they are sensitive to deformations of the extremal black hole beyond the usual addition of mass at fixed charges, using the example of the Reissner-Nordström black hole. We will also describe the gravitational perturbation that takes an extreme Kerr black hole away from extremality and how its dynamics is related to JT gravity in Bondi gauge. In particular we will give an interpretation of the additional U(1) symmetry as the axial symmetry of the rotating black hole. In this section, we will use the Lorentzian conventions for the variables P, T, ϕ 0 , ϕ 1 . The dictionary with the Euclidean variables is given in (2.14).

Deformations of Reissner-Nordström
We start with the near-extreme Reissner-Nordström black hole. It is known to give JT gravity after a Kaluza-Klein reduction on the sphere [84,85]. The 4d geometry is given by The inner and outer horizons are at where M is the mass and Q RN is the electric charge of the black hole. In this section, we use units in which the 4d Newton constant is set to G (4) N = 1. The black hole is extremal when the two horizons coincide where M 0 = Q RN is the extremal mass. The near-extreme black hole is obtained with a deformation where λ is a small parameter, and δr ± are two constants, which can be translated into deformations of the black hole mass and charge. The near-horizon geometry is then obtained by replacing and taking the limit λ → 0, where we used the same parameter λ as in (7.4). In the limit λ → 0, we obtain the AdS 2 × S 2 metric where the metric of the sphere is ds 2 (S 2 ) = M 2 0 dΩ 2 . In FG gauge, we can write the AdS 2 metric as This is the usual AdS 2 geometry (3.1) and we can read 13 which determines the AdS 2 inverse temperature β. The above equation shows that the formulation in FG gauge is only sensitive to the sum of the deformations δr + + δr − .
In contrast, our Bondi-AdS formulation of JT gravity will be sensitive to the independent values of δr + and δr − . For this reason, it can be seen as a finer version of near-AdS 2 holography which differentiates between a larger set of deformations. The additional information will be related to the U(1) symmetry of our AdS 2 boundary action.
To take the near-horizon limit to AdS 2 -Bondi, we consider the Reissner-Nordström black hole in Eddington-Finkelstein coordinates We make the deformation (7.4) and take the near-horizon limit using This leads, after rescaling some of the coordinates, to the AdS 2 × S 2 geometry with We note that these are precisely the geometries that are captured by the Bondi-AdS version of JT gravity. We see that the we are indeed sensitive to the independent values of δr + and δr − . As a consistency check, we can verify that as given in (7.8), and with AdS = M 0 . The usual formulation of JT gravity, in FG gauge, is only sensitive to the combination that gives s 0 , while our formulation distinguishes between different values of P 0 and T 0 .
Thermodynamics. The simplest deformation of an extremal black hole is to add some mass at fixed charge this gives the deformation (7.4) with The entropy of the near-extremal black hole is then where S 0 = πM 2 0 is the extremal entropy. The second piece is the entropy added by the small addition of mass. This entropy can be reproduced using the Schwarzian theory [16], whose entropy is given by S Schwarzian = 2π 2γE . (7.17) This matches with the perturbation of the near-extremal black hole entropy for E = λδM and γ = κφ r = λM 3 0 . It is natural to consider a more general deformation of the two horizons, i.e. to take δr + = δr − . For example, this can be achieved by a deformation of the mass and charge given by This corresponds to the following deformation of the extremal black hole: we first increase both the mass and charge by the same O(λ) amount so that the black hole remains extremal and we then increase the mass by an O(λ 2 ) amount. This regime is necessary to get the effect that we want: a deformed geometry with δr + = δr − . In this more general case, the near-extremal entropy and temperature are S = πM 2 0 + 2πλM 0 δr + + O(λ 2 ) , (7.20) The near-extremal entropy can be written as where the first term is a correction to the extremal entropy taking the form 22) and the second term is linear in the Hawking temperature β −1 H . We expect that JT gravity captures only the term linear in temperature. Indeed, the entropy of JT gravity takes the form where β −1 is the AdS 2 temperature which is related to the Hawking temperature β −1 H according to We can also obtain the values of the charges E and Q introduced in Sec. 4. In the saddle-point approximation, they are given by We note that the additional charge Q allows us to probe deformations with δr + = δr − . Therefore the version of JT gravity in Bondi gauge is a finer probe of near-extremal black holes deformations. As a consistency check, we can further verify that the entropy is the correct linear response in temperature in the near-extremal entropy. It is also interesting to note that our U(1) charge Q corresponds here to the change in the electric U(1) charge Q RN of the black hole, as can be seen from (7.19).

Breaking away from extreme Kerr
The focus of this work is on JT gravity. Nevertheless, the ideas developed here should be applicable in more general near-AdS 2 spacetimes, even in cases where it is not clear if JT gravity is a good description, e.g. when it cannot be obtained by Kaluza-Klein reduction. One such case of interest is the near-horizon geometry of the extreme Kerr black hole (NHEK) [83]. The near-AdS 2 physics of near-extreme Kerr was realized in [59] as a linearized perturbation of its near-horizon geometry, where one of the mode of the perturbation was shown to be the Schwarzian mode. In this section, we perform an identical analysis in Eddington-Finkelstein coordinates, which allows us to obtain the AdS 2 factor in Bondi gauge. We consider a consistent perturbation of the NHEK where one of the mode will satisfy the dilaton equations of motion (2.12). We show how the infinite-dimensional asympotic symmetry algebra of AdS 2 is embedded in the the near-horizon geometry of extreme Kerr. We also see that our additional U(1) charge gets interpreted as the angular momentum of the 4d geometry.

(7.28)
This geometry has an SL(2, R) × U(1) isometry group. It contains an AdS 2 factor in Bondi coordinates. We can consider a more general AdS 2 background as given in (2.6) and (2.7). To do so, one can act on the (u, r)-coordinates with the Lorentzian version of the diffeomorphism (2.28), together with a change in the azimuthal coordinate φ: We notice that the azimuthal coordinate is sensitive to the zero mode of G. This already shows that our U(1) symmetry is related to the change of angular momentum. This diffeomorphism leads to the geometry with general values for P (u) and T (u), written in terms of F and G. To be able to distinguish between these geometries and obtain non-trivial dynamics, we need to deform the geometry away from extremality. This is done by considering a linearized gravitational perturbation which brings us to the near-AdS 2 regime. We consider a similar ansatz than the one used in [59] for the perturbation: where a linearization in λ is implied. The deformation is parametrized by three functions ψ, χ and Φ that depend on u and r, and a gauge field We now solve the 4d linearized Einstein equation The coordinates are denoted x a = (x µ , θ, φ) = (u, r, θ, φ). Let us explain some of the steps to obtain the solution. Firstly, from R rr = − 1 2 ∂ 2 r Φ = 0, we obtain whereφ 0 and ϕ 1 are arbitrary functions of u. The equations R rθ = R θφ = R tθ = 0 allow us to determine the θ-dependence in A u and A r . Then, the equation R θθ = 0 gives the equation where 2 is the Laplacian for the AdS 2 metric ds 2 (AdS 2 ) = g µν dx µ dx ν = 2 − r 2 2 + P (u)r + T (u) du 2 − 2dudr . (7.36) We see that χ is an AdS 2 scalar with conformal dimension ∆ = 2 as was obtained in [59]. From R rφ = R φφ = 0, we can determine A u and A r . Then, R tr = 0 allows us to determine ψ. Finally, from R tt = R tφ = 0, we obtain the JT equations of motion. To see this, we first defineφ 0 (u) = c 0 + ϕ 0 (u) which gives where c 0 arises here as an integration constant. Then, R tt = R tφ = 0 gives We see that Φ JT satisfies the equations of motion of the JT dilaton on the background (7.36), as reported in (2.12). The additional constant c 0 was also observed in [59]. We report the solution for the gauge field where α(u, r) is an arbitrary function arising from the integration and we have defined We also have a relation that determines ψ(u, r) ∂ 2 r ψ(u, r) = 3Φ JT (u, r) . (7.41) which is solved by We can then verify that all the components of the Einstein equation are satisfied. The solution space is now parametrized by α(u, r), ψ 0 (u), ψ 1 (u), ϕ 0 (u), ϕ 1 (u), χ(u, r) and Φ 0 , and their dynamics is controlled by the equations (7.35) and (7.38). The functions α(u, r), ψ 0 (u), ψ 1 (u) are expected to be pure diffeomorphisms in the sense that they can be generated by acting on the unperturbed background with Lie derivatives. This is shown for a similar function α in Appendix B of [59]. The constant c 0 corresponds to the perturbation of the angular momentum induced by the deformation. Indeed, note that the deformed geometry (7.31) possesses also an axial U(1) symmetry, materialized by the Killing ∂ φ . In the AdS 2 factor, this symmetry is associated with a 2d gauge field which appears in the circle fibration and is given by This is an electric field in AdS 2 with leading charge normalized to unity. From the expression of A in (7.39), we see that the effect of c 0 is to change this charge at linear order in λ, which corresponds to a change of the 4d angular momentum. The constant c 0 is also related to the U(1) gravitational charge introduced in Sec. 2.3. To see this, we compute the value of Φ that corresponds to the linearized perturbation to (7.28) that one obtain when considering the next to leading order of the near-horizon limit (7.27). This "vacuum" value of is Φ vac = cste · r. If we now parametrize ϕ 0 and ϕ 1 in terms of f and g as in (2.20) and (2.22) and ask their vacuum values, i.e. f (u) = u and g(u) = 0, to correspond to the vacuum value of Φ, we obtain the relation c 0 + iφ rμ = 0, (7.44) between the constant c 0 and the chemical potential. This relation translate into a relation between the 2d gravitational U(1) charge (2.64) and the constant c 0 Hence, the U(1) symmetry discussed in Sec. 2.3 is realized geometrically in the near-extreme Kerr black hole: it corresponds to the axial symmetry of the deformation. The corresponding charge computes the change in angular momentum of the 4d solution. Although the Killing ∂ φ does not act geometrically on AdS 2 , it has a non-trivial effect on the near-AdS 2 physics. We emphasize again that this effect would be invisible in the Schwarzian theory but is captured in the AdS-Bondi boundary action introduced in this paper.

Flat holography in two dimensions
In this section, we study a flat space version of our boundary action. This action was derived in [52] and shown to correspond to a version of the CGHS model. After a brief study of its solution space and its gravitational charges, we show that it can also be interpreted as a particle moving in a 2d flat spacetime. We compute the exact partition function of the boundary theory and the full gravitational path integral. The latter is greatly simplified by the fact that there are only two flat surfaces with boundaries: the disk and the cylinder. We find a non-vanishing contribution of the cylinder indicating that this theory is holographically dual to an ensemble average.

The CGHS model and its boundary action
The CGHS model [86] is a theory of two-dimensional gravity in asymptotically flat spacetime. Following [87], the authors of [52] define a modified version of the CGHS model, dubbed CGHS, whose action is The corresponding equations of motion are In App. A we comment on the AdS version of this model and show that under appropriate gaugefixing conditions, its dynamics reproduces the one described in Sec. 2. The flat version of the metric (2.68) is ds 2 = 2 (−iP (τ )r + T (τ )) dτ 2 + 2idτ dr . (8.6) where P and T are arbitrary functions. It is the most general flat metric which satisfies R = 0 in Bondi gauge. It is readily obtained as the AdS → ∞ limit of its AdS counterpart.
We will see that the theory CGHS has thermal solutions corresponding to the Rindler spacetime, which we describe explicitly below. In [52] it is noted that the on-shell value of Ψ controls the temperature of the Rindler solution according to The above model is closely related to another version of the CGHS model described by the action where Λ is a "cosmological constant" 14 that is fixed from the start. This theory has the same equations of motion as the theory (8.1) with Ψ = Λ. From the above discussion, we see that this theory has only a thermal solution with temperature β = 2πφ r /Λ. In particular, flat JT gravity (corresponding to Λ = 0) has only the zero temperature solution. 15 Since we would like to obtain a non-trivial Euclidean path integral, the theory should contain solutions with different temperatures. This is why we consider the theory (8.1).
After a choice of appropriate boundary conditions, the gauge field does not lead to additional degrees of freedom. We refer to [52] for a detailed analysis of the solution space, including the derivation of the boundary action This action can also be obtained as the AdS → ∞ limit of our Bondi AdS action (2.23).
The asymptotic symmetry algebra is the same than in AdS, it is spanned by the vectors where ε and σ are periodic functions. When expanded in modes, they satisfy the warped Witt algebra. The corresponding variations of P and T read δ ξ P = εP + ε P + ε , δ ξ T = εT + 2ε T + σ P − σ . where we refer to Sec. 2.2 for more details. Following a similar procedure to the one in Sec. 2.3, we derive the gravitational charges. The computations are given in App. C. In terms of f and g, the gravitational charges read We recall that Ψ is just a constant on-shell. These charges define a centerless representation of the asymptotic symmetry algebra under the modified bracket (2.54). The expression of the nonintegrable part indicates that a proper phase space is achieved when P and T are constants, held fixed in the solution space, which we denote P 0 and T 0 . This condition realizes the symmetry breaking from the infinite-dimensional algebra of asymptotic symmetries to the subset that preserves these values of P and T , i.e. satisfying This is solved by The generators λ 2 and λ 3 are well defined only if P 0 = ±2πi/β (we do not consider winding here). The corresponding algebra is ISO(2)×U(1), a central extension of the 2d Poincaré group. Defining the corresponding generators K i , for i = 1, ..., 4, the only non-vanishing commutators are where K ± = K 2 ± K 3 . Therefore, we see that asking P 0 = ±2πi/β and T = T 0 realizes the symmetry breaking In that case, the charges become integrable and conserved. For the same values of P and T , the boundary action (8.9) is invariant under the same symmetry. The centrally extended Poincaré algebra acts on the fields according to where ε and σ are given by Eq. (8.15). Moreover, the gravitational charges are then identified with the Noether charges of the boundary action.
With the boundary conditions considered in [52], one of the components of the dilaton equation in (8.5) becomesφ Integrating this equation and considering the thermal solution, we obtain the relation (8.7) between the temperature and the value of Ψ. 16 The latter is not fixed by the equations of motion, allowing us to consider solutions with different temperatures. We should make a comment on the fact that asking a maximum of symmetry does not uniquely fix the solution. Indeed we still have the freedom to choose the value of T 0 . A similar freedom is present in AdS-Bondi, where only the combination s 0 = −T 0 − 1 2 P 2 0 was fixed by the temperature, see the end of Sec. 2.3. Again this freedom is materialized by transformations that do not belong to the group, indeed, all values for T 0 are obtained by acting on the couple which corresponds to G(τ ) = β 2πi T 0 τ and F(τ ) = τ , which does not belong to the group because G is not periodic.

A particle moving in flat space
In this section, we will show that the boundary action (8.9) can be interpreted as a particle moving in flat space. Let's consider a particle moving on the background geometry (8.6) for P (τ ) = P 0 and T (τ ) = T 0 . We consider a "vacuum" trajectory which corresponds to a particle lying at r = r 0 . Applying a warped Virasoro transformation (f, g), we obtain a new particle with trajectory The worldline action of the particle is defined as The computation shows that this matches with the boundary action (8.9) up to a constant term As a result, the dynamics of the CGHS model is captured by the motion of this particle. We depict the boundary particles of classical solutions in Fig. 3.
Minkowski particle The vacuum of the theory is 2d Minkowski space where u = −iτ is the Lorentzian Bondi coordinate. This corresponds to the geometry (8.6) with The corresponding boundary particle lies at a constant value r = r 0 . In fact, this solution actually has two boundary particles because there is another asymptotic boundary at r = −r 0 . This geometry is the analog of the global AdS 2 solution of JT gravity studied in [88] which has the interpretation of an eternal traversable wormhole, see Fig. 3.
Rindler particle There is also a thermal solution which corresponds to the Rindler spacetime. This is a 2d analog of the Schwarzschild black hole. To describe this configuration, we can start with the Rindler metric with inverse temperature β where we have t ∼ t + iβ. We can write this in Bondi gauge with the change of variables x = βr π , t = u + β 4π log r , (8.27) which leads to the metric The Euclidean metric is obtained by u = −iτ . This corresponds to the geometry (8.6) with Note that this geometry can be generated by the diffeomorphism (2.28) with the choice We depict the boundary particles corresponding to these two states in Fig. 3.

Euclidean path integral
We consider the Euclidean path integral of the theory (8.1). The integration over Φ fixes the 2d metric to be flat while the integration over Ψ imposes the constraint * dA = 1 on the gauge field.
Assuming that there exists a solution A 0 to the constraint, the general solution is A = A 0 + dλ where λ is an arbitrary scalar field. The integration over A becomes an integration over λ, which needs to be quotiented by the trivial gauge transformations (i.e. the ones that vanish at infinity). The only remaining degree of freedom is the boundary value of λ, which is fixed by the boundary condition considered in [52]. What remains is the path integral over f and g of the boundary action (8.9). In this section, we will consider the only two flat surfaces with asymptotic boundaries, the disk and the cylinder, which will completely determine the gravitational path integral.

Partition function
The partition function is obtained by computing the path integral of the action (8.9) on the disk geometry. The result for the disk partition function can be deduced from a computation done in [77]. We revisit it here, allowing general values of P 0 and T 0 in the Pfaffian, which will be necessary for the computation of the cylinder contribution in the next section. This computation is very similar to the corresponding AdS computation in Sec. 4.2. As in the AdS case, the disk partition function is one-loop exact because of the Duistermaat-Heckman theorem.
To compute the one-loop contribution, we decompose where ε and σ are taken to be infinitesimal. The boundary conditions impose that ε and σ are periodic with period β. Hence, they can be decomposed as Since ε(τ ) is real and σ(τ ) is pure imaginary, we have the relations The symplectic form on the phase space takes the form We can write ω εσ as a 2M × 2M indexed by (n, m) where −M ≤ n, m ≤ M and n, m = 0.
(ω εσ ) nm = α β 2 2πi − 2πi β n 2 + P 0 n δ n+m . The classical piece of the action is given by 38) and the quadratic action is given as where We can perform the Gaussian integral for generic values of P 0 and T 0 and we obtain We now consider the values of P 0 and T 0 that correspond to the Rindler spacetime To compute the one-loop piece, we rewrite the Pfaffian as Pf n (ω), where we have removed the degenerate directions ε 1 and σ −1 and defined Pf ε −1 ,σ 1 = 2αβ. The one-loop path integral can be decomposed as where we compute This leads to Since T 0 = 0, the classical piece (8.38) vanishes here so the partition function takes the form which matches the result of [77]. 17 An inverse Laplace transform gives the density of states Since there are no higher genus surfaces with a flat metric, this is actually the exact density of states. 18 The fact that ρ(E) is not a sum of delta functions implies that our model is not a standard quantum system. In fact, we will see that this theory should be interpreted as an ensemble average. We note that (8.48) is different from the density of states of the CGHS model that was proposed in a different context in [76]. They actually consider the theory (8.8) which has a fixed temperature and is different from the theory (8.1) considered here.

Cylinder
The cylinder is the flat space analog of the double trumpet. It is described by the metric where b > 0 is the circumference of the cylinder and is depicted in Fig. 4.
To compute its contribution, we first compute the contribution of half-cylinder which is described by the same metric, where we only focus on the asymptotic boundary at x = +∞. We can define a Bondi coordinate τ with to get ds 2 = b 2 β 2 dτ 2 + 2idτ dr . (8.51) This corresponds to the geometry (8.6) with  Figure 4: The cylinder geometry. We specify two boundary conditions β 1 and β 2 at each end. The only modulus is the circumference b.
The one-loop contribution is computed in the same way as in the previous section. The difference is that because P 0 = 0, the Pfaffian (8.36) takes the form We note that this is the same contribution as the AdS trumpet for µ = 0.
We can now compute the formula for the cylinder using where the factor b in the measure bdb is necessary because of the freedom to twist one of the half-cylinder relative to the other when gluing. In AdS, this factor follows from the Weil-Peterson measure [30]. This gives Z cyl (β 1 , β 2 ) = 4π 2 γ β 1 + β 2 , (8.58) where we have set α = 1. It is interesting that we find a non-zero answer. This implies that if the CGHS model has a holographic dual, it has to be an ensemble of theories, as in the case for AdS JT gravity. Let us introduce the notation Z(β 1 ) . . . Z(β n ) to represent the Euclidean path integral with n asymptotic circles of lengths β 1 , . . . , β n . The fact that the cylinder does not vanish implies that Z(β 1 )Z(β 2 ) = Z(β 1 ) Z(β 2 ) . (8.59) This indeed shows that the path integral should be interpreted as an ensemble average. The answer (8.58) is not the universal answer for double-scaled matrix ensembles so the dual of the CGHS model has to be something different.

Genus expansion
The Euclidean path integral that we consider involves flat Riemann surfaces with at least one asymptotic boundary. The only such surfaces are the plane (or disk) and the cylinder. Therefore, the connected contribution to the path integral with more than three boundaries identically vanishes: Z(β 1 )Z(β 2 ) . . . Z(β n ) c = 0 n ≥ 3 . This shows that the path integral with an arbitrary number of boundaries is completely determined using Wick contractions involving the cylinder and the disk. This implies that the corresponding third-quantized theory is a Gaussian theory. Thus, the CGHS model constitutes an interesting example of a theory where the full Euclidean path integral can be done, while not being completely trivial and giving rise to an ensemble average.

A Gauge-theoretic formulation
In this paper, we have been focusing on an AdS-Bondi version of JT gravity. It is worth mentioning that there is a closely related theory which gives an alternative formulation of the same physics. This is an AdS version of the CGHS model studied in Sec. 8.1 and described by the action This theory can be interpreted as JT gravity coupled to a BF theory via a scalar field Ψ. The equations of motion are Following [52], we require this solution to be preserved in the phase space, under combined diffeomorphisms and gauge transformations. A similar condition in AdS 2 was considered in [89]. This leads to the same asymptotic symmetry algebra than the one considered in Sec. 2. Indeed asking the metric to be preserved gives ξ = ε(τ )∂ τ − (ε r − η(τ ))∂ r .

B Extrinsic curvature in Bondi gauge
In Sec. 2, we have derived the boundary action for JT gravity in Bondi gauge by requiring the variational problem to be well-posed. In Sec. 3, we have explained how one can recover the Schwarzian action after going to Fefferman-Graham gauge. The Schwarzian action is usually derived from the extrinsic curvature term where h is the boundary metric. We will show here that this term does not give the right boundary action for JT gravity in Bondi gauge.
To compute the extrinsic curvature, we start with the Euclidean metric ds 2 = 2 r 2 2 − iP (τ )r + T (τ ) dτ 2 + 2idτ dr , (B.2) with the value of the dilaton associated to (f, g). We can map this to the constant dilaton 19 by using the diffeomorphismτ such that the above diffeomorphism for (f ,g) is the inverse of the corresponding diffeomorphism for (f, g). In particular, we haveτ = f (f (τ )) = f (τ ). Because we are now in a situation with constant dilaton, we should evaluate the extrinsic curvature on the curve at constant r. We find which can be written using the variable τ as where we have used the expression for s derived in Sec. 3.1 As a result, we see that the extrinsic curvature term does not depend on g(τ ) and matches with the Schwarzian action discussed in Sec. 3. Therefore, the extrinsic curvature does not give the right boundary term in Bondi gauge, but additional counterterms seem to be needed. Alternatively, the approach taken in Sec. 2 provides an efficient and unambiguous way to derive the boundary action.

C Gravitational charges of the CGHS model
We derive here the expression of the gravitational charges given in Sec. 8.1, associated with the warped Witt asymptotic symmetry algebra of 2d Minkowski spacetime. This result follow from boundary conditions on the solution space of the CGHS.