Warped AdS_3 Black Holes in Higher Derivative Gravity Theories

We consider warped AdS_3 black holes in generic higher derivatives gravity theories in 2+1 dimensions. The asymptotic symmetry group of the phase space containing these black holes is the semi-direct product of a centrally extended Virasoro algebra and an affine u(1) Kac-Moody algebra. Previous works have shown that in some specific theories, the entropy of these black holes agrees with a Cardy-like entropy formula derived for warped conformal field theories. In this paper, we show that this entropy matching continues to hold for the most general higher derivative theories of gravity. We also discuss the existence of phase transitions.


Introduction and Summary
Two-dimensional conformal field theories (2d CFTs) and their celebrated Cardy formula play a central role in our attempt at understanding the microscopic degrees of freedom responsible for black hole entropy. Strominger and Vafa's seminal derivation [1] of the entropy of the three-charges extremal D1-D5 black hole revealed that the low energy effective field theory governing the excitations of the system flows in the infrared to a 2d CFT, whose asymptotic number of states counted by the Cardy formula exactly accounts for the Bekenstein-Hawking entropy. This result was eventually generalized to a variety of other setups, including both extremal and near-extremal higher-dimensional black holes [2][3][4][5][6][7][8][9], and generic BTZ black holes [10,11] in three dimensions [12]. The matching between black hole and Cardy entropies could be traced back to the presence of an AdS 3 factor (in the near-horizon region for the higher-dimensional cases) of the corresponding geometries. The result followed as a consequence of the work of Brown and Henneaux [13] who showed that the classical phase space of AdS 3 gravity is endowed with the action of the 2d local conformal group, implying that states of the corresponding quantum gravity theory should organize into representations of a 2d CFT [12].
More precisely, the Cardy formula is a well-known universal property of unitary and modular invariant 2d CFTs encapsulating the asymptotic degeneracy of states at large charges/temperatures for fixed central charges: Remarkably, this formula matches the Bekenstein-Hawking entropy of the BTZ black holes for all values of the charges/potentials (and not only L ± 0 → ∞ or T ± → ∞). The reason for this seemingly unreasonable effectiveness was clarified in [14] (see also [15][16][17] for related earlier works): for CFTs with a large central charge (corresponding to the semi-classical limit on the gravity side) and a sparse light spectrum, the Cardy regime extends all the way down to where the condition L + 0 + L − 0 ≥ c 12 is satisfied. This behavior puts constraints on the precise nature of 2d CFTs potentially dual to AdS 3 theories of gravity, examples of which include symmetric orbifolds [14,[17][18][19][20][21] and extremal CFTs [22,23]. Other CFTs potentially dual to AdS 3 gravity have recently been discussed in [24] The key property responsible for the existence of a simple and elegant formula like (1) is modular invariance.
For a partition function at inverse temperature β, it is expressed by the equality It is the latter expression that allows to express the high temperature partition function using only minimal data about the vacuum of the theory. The universal character of the asymptotic growth of the number of states in 2d CFTs, as captured by the Cardy formula, is a consequence of the peculiar UV/IR connection in 2d CFTs embodied in (3), relating high/low-temperature regimes of the theory. A similar behaviour is expected to explain the universality of the area-entropy law in gravity theories. On the one hand, the Bekenstein-Hawking entropy S BH = Area/4G N captures to leading order in 1/G N the UV density of states of a quantum gravity theory governed by the Einstein action in the IR. On the other hand, it is a reasonable expectation that, beyond its mere existence, the specific details of a UV-complete gravity theory are actually not needed to understand the area law, much like the actual quantum theory of atoms was not needed to derive the universal laws of thermodynamics for gases from statistical reasoning [25].
In any sensible approach to quantum gravity, the Einstein-Hilbert action will only provide the leading low-energy effective action. At high energies, it is expected to be corrected by higher-curvature terms suppressed by a dimensionful scale. Typically in string theories, the coupling would correspond to the inverse string tension α . In that case, the entropyarea law is no longer expected to hold, and is modified in order for the first law of black hole mechanics to hold [26][27][28]. Interestingly, the match between Cardy and black hole entropies still persists [9,[29][30][31][32][33][34][35][36][37][38][39][40]. The effect of higher-curvature corrections in AdS 3 gravity boils down to a global multiplicative renormalization of all the charges of the theory.
Of course, AdS 3 spaces and asymptotically AdS boundary conditions are believed to be part of just one (although possibly the simplest) among many manifestations of the holographic principle, and intensive work has been devoted to depart from the AdS/CFT paradigm. This includes for instance dS/CFT [41], AdS/Condensed Matter Theory [42,43], Non-relativistic Schrödinger and Lifshitz Holography [44], Hyperscaling geometries [45], Flat Space Holography [46] and Kerr/CFT [25,47] (references are by no means meant to be extensive, and point most of the time at reviews on the subjects).
In this work, we will focus on one particular example of non-AdS Holography: Warped AdS 3 spaces (WAdS 3 ) and Warped Conformal field Theories (WCFTs) [48][49][50][51][52]. The former have appeared in various contexts, in relation to Kerr/CFT and Schrödinger holography, as UV deformations of AdS 3 spaces. They are parameterized by two constants which are usually taken as , the original AdS 3 radius, and (ν 2 − 1) characterizing the departure from AdS 3 . They were observed to display an Asymptotic Symmetry Group (ASG) different from the Brown-Henneaux conformal symmetry group, consisting instead in the semi-direct product of a Virasoro algebra and a u(1) affine Kac-Moody algebra [53][54][55][56][57]. WCFTs are defined as two-dimensional field theories with precisely these symmetries. Interestingly, these theories possess an infinite-dimensional symmetry group as well as a notion of modular invariance allowing the derivation of a Cardy-type formula for the density of states given by [51] S WCFT = −4πi where L 0 and P 0 are the zero-modes of the Virasoro and Kac-Moody (KM) current generators respectively, β the inverse temperature and Ω the angular velocity, c is the Virasoro central charge and k the u(1) level. P vac 0 is the charge of the vacuum state on the cylinder which is not fixed by the symmetries of the theory through a plane-to-cylinder map like in 2d CFTs 1 . Further properties of these new classes of field theories have been explored e.g.
So far, the matching between Bekenstein-Hawking and field theory entropies for WAdS 3 black holes has been performed on a case-by-case basis in specific theories admitting WAdS 3 spaces as a solution -these backgrounds do not solve the vacuum Einstein's equation, so they require including matter fields or higher curvature corrections (see e.g. [51] for Topologically Massive Gravity [67,68] and string theory embeddings, [69,70] for New Massive Gravity [71], [72,73] for Born-Infeld extensions of NMG). The goal of the present work will be to extend this to a completely arbitrary higher-curvature gravity theory admitting WAdS 3 as a solution. We will indeed show that the black hole entropy is always reproduced by (4) or (5). The result is comparable in spirit to the analogous statement for BTZ black holes that their entropy is always reproduced by a Cardy formula [29,39], and similarly in Kerr/CFT where the original derivation of [74] could be extended to an arbitrary higher-curvature 1 Eq. (1) also implicitly contains the vacuum charges on the cylinder, but in a unitary CFT these are given by L ±,vac 0 = − c ± 24 . In WCFTs, only L vac 0 is constrained by unitarity bounds to be L vac theory [75]. We will closely follow the philosophy of the latter work. On the one hand, we will compute the black hole entropy of the WAdS 3 black holes given a general gravity theory with a Lagrangian L. It is known that in the presence of higher-derivative corrections, the entropy is no longer given by the horizon area, but rather by Wald formula [26][27][28]76] which in 3d reads as where Z αβµν is constructed out of curvature invariants of the metric and possesses the symmetries of the Riemann tensor and αβ is the binormal at the horizon r + . In the case where the Lagrangian is only a function of the Riemann (without covariant derivatives), then symbolically Z ∼ ∂L/∂Riemann. In general, the expression is more complicated (see Sec. 3 for details).
Applying Eq. (6) to the WAdS 3 black holes, since all these black holes have a local SL(2, R) × U (1) isometry, this will allow us to rewrite the Wald entropy in a very simple form as where A is a constant depending only on (ν, ) and the couplings of the theory (in particular, they do not depend on the black hole parameters). Then, exploiting again the symmetries and using the Covariant Phase Formalism [26,27,75,[77][78][79][80] we will be able to compute the charges and central charges appearing in (4) and (5). In particular, we will find that which, when plugged in (5), exactly reproduces (7): In the body of the paper we will further discuss equality in other ensembles.
As a by-product of our computations, which allowed us to compute the charges of the Warped Black Holes in arbitrary theories, we reconsider the existence of a Hawking-Page phase transition for these backgrounds, generalizing the results of [81] in Topologically Massive Gravity.
The paper is organized as follows. We will first review in Sect. 2 the phase space and asymptotic symmetry group of asymptotically WAdS 3 spacetimes as well as the WCFT entropy formula in WAdS/WCFT holography. The warped black hole entropy in New Massive Gravity will be provided to review some known results of the entropy matching.
Moving on, in Sect. 3.1, we further investigate the geometry of WAdS 3 and argue that all curvature invariants can be expressed in a manner that will allow us to derive, in Sect 3.2, the Wald entropy of the warped black hole in a very concise form (see Eq. (7)). Moreover, these simplifications will make possible the computation of the charges for an arbitrary theory in Sect. 4

Phase Space
WAdS 3 spaces [48,50,82] can be obtained by deforming the AdS 3 metric in the Kerr-Schild-like manner, breaking the SL(2, R) × SL(2, R) isometry group of AdS 3 down to a SL(2, R) × U (1) isometry group. The metric in global coordinates is in which we recognize the AdS 3 metric (for ν = 1) but otherwise with a squashing/stretching factor 4ν 2 /(ν 2 + 3). In this case, black hole solutions exist and are quotients of the WAdS 3 metric. The black hole metric can be written in the so-called warped-black-hole coordinates of [55] as: with r ∈ [0, ∞), t ∈ (−∞, ∞), φ ∼ φ + 2π, and m and j are parameters characterizing the black hole. That metric is a black hole with two horizons in the case where j < 9 m 2 ν 3+ν 2 .
They are located at When the warp parameter ν 2 > 1, the solution is said to be stretched, and when ν 2 < 1, it is squashed. For ν 2 = 1 we recover locally AdS 3 space. We will focus on the spacelike stretched case, which exhibits no pathologies such as naked closed timelike curves, unlike its squashed counterpart.
An important feature of these metrics is that they are part of exact string theory backgrounds, like AdS 3 spaces. The latter are known to be the target space of an exact Conformal Field Theory, an SL(2, R) WZW model [83][84][85][86][87][88]. These models provide exact solutions to all orders in α modulo a renormalization of the WZW level, identified with the AdS radius [89]. WAdS 3 spaces, on the other hand, are part of backgrounds representing marginal deformations of the SL(2, R) WZW model [90]. It was shown in Sect 2.3 of [91] that the background fields extracted from the classical action get renormalized only through a redefinition of the parameters in the metric (the AdS radius and squashing parameter).
Therefore, when considering WAdS 3 black holes in higher-curvature theories, we will have to consider the modification to their entropy only due to replacing the area law by Wald's entropy (and not to a modification of the geometry).

Asymptotic Symmetries
WAdS 3 spaces are not asymptotically AdS 3 (except of course for ν 2 = 1) and do not belong to the Brown-Henneaux phase space [13]. Instead, they satisfy the following boundary conditions [53][54][55][56][57] (coordinates are (t, r, φ)): The infinitesimal diffeomorphisms leaving these boundary conditions invariant are generated by the asymptotic Killing vectors [55] These generators obey the following Lie-commutation relations: The conserved charges L m , P m associated to these generators l n , p n satisfy a Virasoro-Kac- The explicit definitions and expressions of charges (in particular the central charge and the level) and their commutators will be provided later, since they are (bulk)-theory-dependent.

WCFT and BH Entropies in NMG
The above algebra is the defining symmetry algebra of a WCFT, a two-dimensional field theory with global SL(2, R) × U (1) invariance [51,52]. These theories share many features with 2d CFTs, in particular the existence of a formula counting the degeneracy of states at large potentials/charges, given in the micro-canonical ensemble by [51] This formula has allowed to match entropies on the gravity and field theory sides in various theories. We briefly review here the case of NMG [69].
The NMG action is given by where p has the dimensions of a mass square. Warped black holes are solutions of NMG for the following couplings: In terms of the angular velocity the entropy can be expressed as on the gravity side. On the field theory side, the central charge c and level k in the asymptotic symmetries algebra are The WCFT formula for the entropy can be rewritten in terms of Ω and the inverse temperature β as with P 0 is the charge associated to ∂ t and L 0 the one associated to ∂ φ . The vacuum is obtained by taking m = i/6 and j = 0, corresponding to the only metric in the family (13) with enhanced SL(2, R) × U (1) isometry and no conical defect in appropriate coordinates (for a detailed discussion, see [51]), which yields We then have as expected.
The goal of this paper is to show that this matching holds for any higher-derivative gravity theory.

Symmetries and Curvature Tensors
In this section, we summarize some key facts related to curvature invariants constructed out of a WAdS 3 metric. In particular, these results are applicable to the warped black holes since they are quotients of the WAdS 3 metric.
First of all, in 3d, all Riemann tensors can be expressed in terms of Ricci tensors. In the case of WAdS 3 , from [50], we know that some products of Ricci tensors are This can be easily verified for the metric Eq. (13). Furthermore, recall that in a maximally symmetric spacetime, all curvature tensors (for e.g. product of covariant derivatives of Riemann/Ricci tensors) are expressible (covariantly) in terms of the metric tensor. 2 In the present case where we only have SL(2, R) × U (1), a slight generalization of such an expression is available as well. This is in essence the argument in Appendix D of [75].
Any tensor constructed out of the metric should respect the SL(2, R) × U (1) isometry.
The consequences have been investigated and exploited in Ref. [75]. In particular, any scalar curvature invariants constructed out of the metric are constants. As such, in our case, these constants can only depend on ν and and not on the parameters of the black holes (i.e. m and j), which are parameters of the global quotients 3 .
For a general tensor, let us consider what happens in the case of a symmetric-two tensor S µν . Due to boost-invariance (which is a consequence of the SL(2, R) × U (1) symmetry), in a conveniently chosen vielbein basis e0, e1 and e2 (we wrote these as one-forms eî ≡ eî µ dx µ ), which is given explicitly in [75], we have while S22 is arbitrary. This implies that any such tensor only contains two arbitrary components. In particular, we can conveniently decompose it as where the constants c i 's only depend on ν and . The vector J µ is most usefully chosen to where we work in the warped black hole coordinates in which Eq. (13) is written and p 0 is the U (1) R Killing vector (see Eq. (16)). Note that J µ J µ = 1. Translating back into spacetime indices, we obtain Furthermore, note that where the convention is trφ = √ −g = 1. Thus, all products of covariant derivatives of S can in turn be rewritten as products of g, and J. Let us give an example where S µν = R µν . Then, Throughout this paper, we will be interested in a tensor Z αβµν which has the same index-symmetry as the Riemann tensor and is constructed out of the metric (and its derivatives). It also depends on the theory one considers. By general SL(2, R) × U (1) symmetric arguments, we can repeat the above arguments to obtain for some constants A and B that only depend on (ν, ). In this case, we prefer to reexpress J µ J ν in terms of R µν such that the RHS is expressed in terms of product of g µν and R µν .
On the other hand, the equations of motion always take the form where K µν [g αβ ] for a given symmetric-two-tensor constructed out of the metric. For e.g.
in pure Einstein theory, it is just the Einstein tensor or the Ricci tensor. Following the above logic, evaluating K µν on a warped AdS or black hole solution, the symmetry of the geometry allows us to decompose K µν into a sum of the metric and the Ricci tensor: 4 where E 1 and E 2 are constants which only depend on (ν, ) and on the couplings α i of the theories. Note that the dependence on the couplings of the theories is linear. The equations of motion then reduce to two independent equations, setting This means that as long as we have a theory with two independent couplings, such as that in TMG or NMG, then we will always be able to solve the equations of motion. The couplings appear linearly in the E i 's, for that reason we can always solve these two decoupled equations (subject to obtaining real and ν as solutions).

Wald Entropy of Warped Black Holes
For any diffeomorphism covariant theory of gravity, the Wald entropy formula in 3d is where and µν is the binormal at the horizon, given by µν = ∇ µ ξ ν with ξ the generator of the horizon with its surface gravity normalized to unity [92].
In general, the expression of the Z tensor is rather complicated. However, we have seen that by general SL(2, R) × U (1) symmetry arguments (see Sec. 3.1), it can be written as (38). Given Z of the form above, we can compute where we have used µν µν = −2 and defined n µ ν ≡ − αµ αν . Furthermore, using and √ g φφ | r=r + = −Ω −1 , one is led to or 4 Charges in Warped Black holes

Covariant Phase Space Formalism
The definition of (asymptotically) conserved charges is a subtle procedure that can be approached by many differents roads. We will briefly review below the salient features of the formalisms we will be using throughout this work, and refer the reader to the literature for a more detailed summary (see e.g. Sections 3 of [93] or [75], App. A of [94]).
From the Lagrangian n-form L (with n the spacetime dimension), which is a local functional of all fields denoted collectively Φ, the equations of motion E(Φ) = 0 are determined as follows: where Θ[δφ, φ], the symplectic potential (n − 1)-form, can always be chosen to be covariant [27]. We use a slight abuse of notation, writing E(Φ) for both an n-form and its Hodge dual.
We will be concerned here with the situation where Φ only consists in the metric tensor g. In that case, the gauge symmetries of the theory are transformations δ ξ g = L ξ g under which the transformation of the Lagrangian is computed as Under such a gauge transformation, one can rewrite, using Bianchi identities, More explicitly, in a pure gravity theory one has E(g) the last term vanishes by virtue of Bianchi identities, and hence in that case 5 S µ ξ = 2ξ ν E µν . S enjoys the important property that it vanishes on-shell, and hence is called weakly vanishing Noether current. On the other hand, writing (48) for a gauge transformation (δ = δ ξ ) and using (49), one gets where the canonical Noether current is defined as From (50) and (51), one therefore gets that the (n − 1)-form S ξ (E(Φ), Φ) + J ξ (Φ) is off-shell closed and thus exact, and there exists an (n − 2)-form Q ξ (Φ) such that where ≈ stands for on-shell equality. Q ξ (Φ) is the Noether charge as defined by Wald [26], which is not to be confused with the conserved charge generating the action of the symmetry generator ξ on the covariant phase space (denoted H ξ or H ξ for its Hodge dual henceforth).
An important observation is that the definitions in (54) are ambiguous up to the redef- for an arbitrary E(δ ξ Φ, δΦ) anti-symmetric in δ ξ Φ and δΦ. This ambiguity generalizes the one in the symplectic potential (n − 1)-form under Θ → Θ + dY, and hence in the symplectic structure. One proposal to fix this ambiguity [78,79] is by acting on the weakly vanishing Noether current with a contracting homotopy operator, yielding an (n − 2)-form denoted k BB ξ (δΦ, Φ). In essence, this operator is the inverse of the exterior derivative d (see e.g. [95] for an explicit expression). One advantage of this procedure is that it provides a definition of charges depending only on the equations of motion of the Lagrangian, and not on boundary terms. We then have in which the expression of E(δ ξ Φ, δΦ) is known explicitly (see e.g. (3.7) of [75]). Remark that this ambiguity is not relevant for exact symmetries, having δ ξ Φ = 0, but may yield distinct results in the asymptotic context (see [75] for one such example in Kerr/CFT). In this work, we will be working with Iyer-Wald charges, and explicitly check that the extra term E does not contribute.
Finally, the algebra of charges can be represented by a Dirac bracket as follows: This is valid on-shell when the charges are integrable. The second term on the right-hand side is recognized as a central extension, which cannot be absorbed in a redefinition of the generators.
In these expressions, the Z abcd , R abcd , Z abcd|e 1 ...es and R abcd|e 1 ...es are auxiliary fields in terms of which our original Lagrangian can be rewritten without derivatives higher than second order (we will not need their explicit expressions, see however Sect. 4 of [75] for details).
Furthermore, the relevant E term for our computations is [75] E a 3 ...an [£ ξ 1 g; This term will never contribute to the charges. Obviously, it is true for exact Killing vectors.
For the central charge, we take ξ 1 = l n and ξ 2 = l −n (or t n ans t −n for the level) and use the expression (38) of Z abcd . As we will integrate over a t, r = cst surface, we only need to consider the φ component of E. A direct computation shows that it is indeed zero.

Expressions of charges
First, we will derive the expressions of the charges for the case of a Lagrangian without derivatives of the Ricci. The exact charges depend only on Q (0) ξ and Θ (0) and are given by [27] where the integral is (n − 2)-dimensional sphere (t, r=constant) at spatial infinity. There is no term with i ∂φ Θ in δL 0 because ∂ φ is assumed to be tangent to this sphere.
Using the general form of Z abcd (38), an explicit computation leads to where A, B are the constants entering in (38) and depending on the theory.
For the central terms in (59), we note that it is sufficient to consider the terms proportional to n and n 3 for the computation of the level and the central charge, respectively. We In whole generality, the charges associated with exact Killing vectors of a metric satisfying the equations of motion of a given theory are finite. Therefore, we see from (67) that a theory admitting WAdS 3 as solutions must have its constants A and B related in the following way: In other words, the coupling constants of the considered theory should satisfy the above relation. As expected it turns out it is equivalent to satisfy the equations of motion (see details in Appendix A).
Using (69), the charges have the following expression with only one constant depending on the considered theory. The Iyer-Wald entropy becomes A Ω (72) which is the result advertised in Eq. (7).
Note that the expressions of the charges are proportional to their value in NMG, the simplest theory where warped black holes are solutions:

Warped Cardy formula
In this section, we show that the Iyer-Wald and Warped Cardy entropies match. The computation of the latter needs us to evaluate the exact charges for the vacuum solution (m = i/6 and j = 0), which are then given by We substitute these into the WCFT formula and get precisely the Iyer-Wald entropy (72):

Lagrangians with Derivatives of the Ricci
Now we focus on the general case of a Lagrangian with derivatives of the Ricci. The first change with respect to the case without derivative is the appearance of new terms in Z abcd as given by (43). However, thanks to the symmetries, it will keep the same form (38) provided we replace A and B in (38) by some otherÃ andB. In addition, the charges Θ (s) and Q (s) in (60) are expected to be corrected as well. Nevertheless, we show using the SL(2, R) × U (1) symmetry (see Appendix B) that P 0 is not corrected and that L with a 2 , a 3 constants depending on ν, . Thus the charges are (67), (68) with A →Ã and B →B plus the corrections (77). In order to have finite exact charges, the divergences must cancel. It implies The purpose of the first equation is to cancel the r−divergence. To get rid of the r 2 −divergence, the second term in (67) for L 0 should cancel against the first term in (77). Imposing that condition, we get the same expression for the exact charges we had in the case without derivative. In consequence, the Warped Cardy formula again matches with the Iyer-Wald entropy. However, finiteness of the Virasoro central charge requires a 3 = 0. This is a necessary condition in order to have a well-defined phase space with Vir + u(1) Kac-Moody symmetries. We will take this as a supplementary condition on the coupling constants of the theory (though it is not excluded that this condition could derive from requiring the metric to satisfy the equations of motion). In this case, all the corrections vanish. Relation (69) then holds betweenÃ andB and we get the same charges (70) as in the case without derivative of the Ricci.
In conclusion, we have shown that the Iyer-Wald entropy is always reproduced by a Warped Cardy formula using the SL(2, R) × U (1) symmetries and on-shell conditions. We have also found a general explicit expression for the charges.

WAdS 3 Entropy in Higher Curvature Theories: Examples
In this section, we present a few examples with detailed computations.

L
We first deal with the most general Lagrangian involving no derivatives of the Ricci tensor and up to cubic order [96]. As there is no derivative of the Ricci in that Lagrangian, the tensor Z is only proportional to ∂/∂Riemann. We have to consider the derivative of the Ricci scalar and Ricci tensor. After symmetrization, we get δR δR abcd = 1 2 g bd g ac − g ad g bc (79) So, Meaning that the constants A and B are The charges obtained by (67) and (68) are consistent with their expressions found by other techniques.

Corrections for a Theory with (∇ a R bc ) 2
Take a Lagrangian with the higher curvature term Following [75], we can compute the Z-terms entering the charges: which can be rewritten as (38), using (119): −g ac g bd + g bc g ad − g bd g ac + g ad g bc and = 1 2 g ac ∇ e R bd − g bc ∇ e R ad + g bd ∇ e R ac − g ad ∇ e R bc (90) and the relevant corrections to Θ and Q are Using the relations (36), (37), and δR bc = ν 2 −3 2 δg bc , together with the decomposition of the Riemann tensor, we get the form given by (134) in the Appendix: We can then rewrite the corrections as Since we are going to integrate over φ, and we have to contract the correction to Θ with ∂ t , we need to compute Θ (1) tφ and (Q ξ ) (1) φ . We also know that δg only non-zero components are δg rr and δg φφ . It is then straightforward to show that Even if (Q ∂t ) φ is non-zero, the contribution of this term to the charge vanishes when we take the δ of it. The same results are recovered using the general method proposed in [75].

Phase Transitions
To study possible phase transitions between black holes and thermal states, one must first identify the ensemble black holes are dual to. As argued in Sect 2. of [81] or Sect 5.3 of [51], WAdS 3 black holes are dual to the ensemble The coordinates T and Φ appearing in (99) are natural coordinates when expressing WAdS 3 black holes as a deformation of BTZ black holes 6 :: where ds 2 BT Z is the BTZ black hole metric (where we put Newton's constant G to 1): and ξ such that ||ξ|| 2 = 1 is given by For the change of coordinates between (13) and (101), see e.g. Sect 5.4 of [51].
The symmetry argument is still valid in that coordinate system and so the Z abcd field has the same form (38). Doing similar computations, the exact charges are given by with C a constant depending on the coupling constants, on the deformation parameter H 2 and b . For example, NMG has The thermodynamic potentials are given for any theory by Integrating the first law, the Gibbs free energy is given by (108) 6 The deformation parameter H is given in terms of ν by H 2 = 3(1−ν 2 ) 2(3+ν 2 ) , and b = 2  First, we study the local stability of these phases. In the grand canonical ensemble, the stability condition is the requirement for a system to have a negative semi-definite Hessian of its free energy G(T, Ω) . This implies that For example in NMG, this condition implies that H 2 < 17/42.
Secondly, we consider the global stability. In the classical limit, the dominant phase is the most probable, i.e. the one who dominates the partition function among the saddle points. Here the two known phases are the black hole and the thermal vacuum. We compare their free energies through their difference If ∆G < 0, WAdS dominates and for ∆G > 0, WBTZ dominates. The constant C factorizes out 8 . It implies that the phase diagram does not depend on which theory we look at and moreover, as thermodynamic potentials also do not depend on the deformation parameter, it is the same phase diagram as for BTZ black holes. The phase diagram is represented (for = 1) in figure 1. Our results differ from those of [97] in which the authors dealt with the NMG case, but considered a different ensemble.

Acknowledgements
We would like to thank Tatsuo  A On-shell Conditions for Theories without Derivatives of

Riemanns
In this section, we shall study the consequences of the equations of motion for the most general theory without derivatives of Riemanns. Moreover, we will show that on-shell conditions imply Eq. (69). Due to the fact that we are in three dimensions, the most general action (without derivatives of Riemanns) is of the form The equation of motion is easily derived (see [98] for example) to be where The object of interest Z µναβ in three dimensions is which is related to P µν by Z µναβ = 1 2 g µα g βν − g αν g βµ + 1 4 g µα P βν − g να P βµ + g βν P αµ − g βµ P αν .
For later purposes, it is useful to note that for a locally WAdS 3 spacetime, due to SL(2, R) × U (1) symmetry, we have that for q ≥ 1, with A q and B q constants which are dependent on and ν. For example, By definition, the A q and B q satisfy the following recursion relations As a notational convention, we will denote tr(R q αβ ) ≡ (R q αβ ) µ µ . Moreover, it is also useful to note that and so while ∇ α R αβ = 0 using the contracted Bianchi identity and the fact that R is a constant.
For illustrative purposes, let us first consider the simple case where for some fixed k ≥ 2, In this case, which upon using Eq. (116) and Eq. (119)-(120) yields By explicitly plugging in the metric and R µν for a locally WAdS 3 metric, these equations in turn become two decouple equations Let us now recall from Eq. (115), Eq. (116) and Eq. (122) that in this case we have Their ratio is where we have used one of the equations of motion Eq. (126). Using Eq. (118), we obtain which is Eq. (69) as required by finiteness of charges.
The upshot is that eventually similar arguments as above follow through and imply Eq. (69) as desired.
where B ija is a tensor symmetric in the (i, j) indices while A kab is antisymmetric in (a, b).
Both A and B are covariant tensors constructed out of the metric. By SL(2, R) × U (1) symmetry, in the vielbein-analysis, we know that by boost-invariance, there are four independent non-zero components for B ija while there are three for A kab . This means that we can decompose them in the following way where a 1 , a 2 , a 3 are constants depending only on ν and .
Let us first focus on the expression for Θ (s) . Using the fact that the only non-zero components of δg ij are the δg rr and δg φφ components and that g ij δg ij = 0, we get Θ (s) a 2 ...an = (δg ij ) b 2 J i g ja + b 3 J i jak J k + (i ↔ j) + b 4 J i J j J a aa 2 ...an .
Moreover J µ ∂ µ = ∂ t implies that the first four terms in the bracket of Eq. (135) (proportional to either δg ij J i or δg ij J j ) do not contribute to Θ (s) . Furthermore, since we are interested in the a = r component (to compute charges at r=constant surface), the last term does not contribute either. Hence, we have established that in the computation of δL 0 and δP 0 , Θ (s) vanishes.
Next, consider the expression for (Q ξ ) (s) For δP 0 , we consider ξ µ ∂ µ = ∂ t = J µ ∂ µ . We see that the second and third term in the square bracket in Eq. (136) vanish since they are proportional to ab J a J b = 0. Thus we are left to evaluate 9 There is a term in the expression of Θ (s) that looks like ZδR which seems like it cannot be manifestly where we have used J µ J µ = 1 and J µ dx µ = dt − (2rν/ )dφ while setting the (a, b) = (t, r).
We should still take the δ of that expression, so it gives us zero.