Warped flatland

We study warped flat geometries in three-dimensional topologically massive gravity. They are quotients of global warped flat spacetime, whose isometries are given by the 2-dimensional centrally extended Poincaré algebra. The latter can be obtained as a certain scaling limit of Warped AdS3 space with a positive cosmological constant. We discuss the causal structure of the resulting spacetimes using projection diagrams. We study their charges and thermodynamics, together with asymptotic Killing vectors preserving a consistent set of boundary conditions including them. The asymptotic symmetry group is given by a Warped CFT algebra, with a vanishing current level. A generalization of the derivation of the Warped CFT Cardy formula applies in this case, reproducing the entropy of the warped flat cosmological spacetimes.


Introduction
The identification of the microscopic origin of gravitational entropy is one of the most fertile puzzles in modern theoretical physics.Progress in this context has been especially significant when it involves the identification of black hole microstates, however, the same issue for cosmological horizons has remained more elusive so far.In the former case, an important breakthrough appeared in the seminal derivation of the microscopic entropy of five-dimensional extremal BPS black holes in the context of string theory by Strominger and Vafa [1].This result has pointed at the central role played by two-dimensional conformal symmetry as underlying the asymptotic or near-horizon dynamics of various classes of black holes [2][3][4][5].This importance has become even more ubiquitous in recent years, when it was observed that 2d conformal symmetry could pertain to the description of generic classes of black holes, including real-world ones [6][7][8][9][10][11][12][13].In addition, beautiful connections have recently been established between the asymptotic symmetries group of 4d Minkowski space, the BMS 4 group [14][15][16][17] and soft gravitons theorems [18][19][20][21], suggesting that asymptotically flat quantum gravity in (3+1) dimensions exhibits a 2d conformal symmetry [22][23][24][25][26].In the case of cosmological horizons, important advances could be made in the context of the dS/CFT correspondence [27,28], also featuring 2d conformal symmetry but many questions are left open [29].
For definiteness and simplicity, we will focus here on gravity in (2+1) dimensions.Its simplest occurrence, pure Einstein-Hilbert gravity, has proven to be a very rich toy model to address numerous questions, providing a simple setup for the gauge/gravity correspondence [2,[30][31][32][33][34][35][36][37].The archetypical scenario consists of AdS 3 gravity with Brown-Henneaux boundary conditions, in which case the corresponding phase space exhibits a two-dimensional conformal symmetry with specific central extensions [2] and contains black holes solutions (the BTZ black holes [38,39]); the entropy of which can be accounted for by the Cardy formula [3].This is summarized in the first column of Table 1.It is worth mentioning that even in the canonical setup of AdS 3 gravity, a wealth of alternative boundary conditions has appeared over the years.These could either consist out of a relaxation of the original fall-offs or the presence of higher-order equations of motion or matter fields, resulting in either the same asymptotic symmetry algebra (ASA) [40][41][42][43][44][45][46][47][48], or in boundary conditions with different ASAs [49][50][51][52][53].
Various new developments have appeared in the last decade exploring departures from the conformal comfort zone.Spaces with non-(A)dS asymptotics have started to draw attention, including Schrödinger or Lifshitz spacetimes relevant to AdS/CMT [54][55][56], Warped (A)dS 3 spaces (W(A)dS 3 ) [57][58][59][60][61][62][63][64], near-horizon geometries of non-extremal black holes [51,[65][66][67][68], or flat space [17,[69][70][71].Central to these approaches is the determination of consistent boundary conditions -which defines the phase space of the theory at hand -and the symmetries preserving them (forming the ASA), hinting at the structure of the corresponding quantum Hilbert space, as in AdS 3 /CFT 2 .In particular, two scenarios paralleling AdS 3 gravity have appeared in the last years: flat space holography in 2+1 dimensions, and WAdS 3 gravity.In both cases, the ASA does not display conformal symmetry.In the former case it consists in the three-dimensional Bondi-Sachs-Metzner-Van der Burg algebra (BMS 3 ), in the latter in the semi-direct product of a Virasoro and a û(1) Kac-Moody algebra.
These observations have hinted at the existence of new classes of two-dimensional field theories (respectively dubbed BMS field theories and Warped Conformal Field Theories or WCFTs [72,73]) as being to Mink 2+1 and WAdS 3 spaces what two-dimensional CFTs are to AdS 3 .Efforts have been devoted in recent years in defining and exploring the properties of such theories, as well as exploiting them in a holographic context.One key point is that these theories, besides displaying an infinite-dimensional symmetry, enjoy a version of modular invariance allowing to single out regimes in which the density of states is captured L0,±1, M0,±1 ∼ iso(2, 1) ⊂ bms3 Subalgebra Field Theory x ± → f ± (x ± ) x − → f (x − ) x − → f (x − ) Symmetries x + → x + + g(x − ) x + → f (x − )x + + g(x − ) Solutions BTZ Black holes [39] Warped AdS3 Black holes (WBH) [77,78] Flat space cosmologies (FSC) [  by Cardy-like formulas [73][74][75].In WAdS 3 gravity, the Bekenstein-Hawking entropy of the so-called spacelike warped black holes [76][77][78] could then be reproduced by the counting of an asymptotic growth of states in a WCFT [73].Three-dimensional pure gravity with a vanishing cosmological constant, on the other hand, notoriously does not contain black hole solutions 1 , but the flat limit of BTZ black holes, called Flat Space Cosmologies (FSC) [79] do enjoy interesting thermal properties and are endowed with a cosmological horizon and entropy.The latter can be matched to a BMS-Cardy formula counting the growth of states in a BMS-field theory [74,75].This is summarized in the second and third column of Table 1.These three scenarios do not stand on an equal footing.The first has a long history, starting in the mid-eighties both on its constitutive parts (AdS spaces and CFTs) and on their close relationship (with the original proposal of [80,81], see [82] for recent advances in deriving the correspondence).In the two latter cases, the field theories are far less understood, but they have attracted considerable attention in recent years.Their intrinsic properties, as well as explicit realizations, have been studied from various perspectives [8,72,73,.Besides the scenarios of Table 1, other holographic realizations featuring non-conformal (and actually, non-Lorentzian) field theories exist of two types: (i) Field theories with Lifshitz-type or more general anisotropic scalings [104][105][106], (ii) Irrelevant deformations of 2d CFTs [107][108][109], relevant to the holographic description of classes of three-dimensional black strings [110][111][112].
In the present work, we will analyze a class of three-dimensional spacetimes that does not fit in any of the above scenarios.This is referred to as Warped Flat (WF) spaces [63,113] and can be viewed as a fibration over two-dimensional flat space along a fiber coordinate spanning the real line.They can be obtained as a scaling limit of WAdS 3 or WdS 3 spacetimes, where a certain flat limit of the (A)dS 2 basis is taken.Global identifications of WF were discussed in [113] in the context of Topologically Massive Gravity [114,115] coupled to Maxwell theory with an electromagnetic Chern-Simons term.The so-called self-dual WF quotient appears as the near-horizon geometry at fixed polar angle of the ultra-cold limit of Kerr-dS black holes where the inner, outer and cosmological horizons coincide [116], much like self-dual spacelike WAdS 3 appears in the near-horizon limit of extremal Kerr [8,91].Another type of quotient was shown to result in a two-parameter family of spacetimes exhibiting a Killing horizon and argued to describe causally regular black holes [63,113].They will be the subject of this work.Black holes in three dimensions [39,43,60,78,[117][118][119][120][121][122] have always provided insightful tools into their higher-dimensional cousins.The existence of asymptotically flat black holes in 3d would in particular allow to explore the role of BMS symmetries in that context, with the simplification that the BMS 3 algebra has a simpler structure than its 4-dimensional counterpart (partly because it can be obtained as a contraction of the two-dimensional conformal algebra).The solutions we will be interested in have a non-trivial Bekenstein-Hawking entropy which, when expressed in terms of the solutions global charges denoted suggestively by P 0 and L 0 , takes the functional form where α 1,2 are constants (see (3.33) for the precise expression).This dependence of the Bekenstein-Hawking entropy on the charges of the solutions is not of any of the three types that can be seen in Table 1.The goal of this paper will be to find a set of boundary conditions accommodating these quotients of Warped Flat space (whose geometric structure will be carefully analyzed), determine their symmetries, and analyze whether those could be used to reproduce the entropy (1.1).The paper is organized as follows.In section 2, we introduce various aspects the geometries we are interested in, in particular their unusual isometries.We analyze their causal structure and conclude that they do not describe black holes, but rather cosmological spacetimes for a certain range of the parameters.In section 3, we embed the solutions in a dynamical theory, which we choose to be Topologically Massive Gravity in Chern-Simons-like form [123][124][125].We work out a set of on-shell boundary conditions including the geometries of interest and determine their asymptotic symmetries.We also compute the thermodynamic quantities of interest, such as mass, angular momentum, Hawking temperature, and Bekenstein-Hawking entropy.In section 4 we turn to a field theory analysis of the symmetries found in the previous section.In particular, following the analysis of [66,73], we derive a Cardy-type formula for a centerless WCFT, which we show matches the geometric entropy previously obtained.Appendix A discusses the limit from spacelike W(A)dS 3 to WF spacetimes.Appendix B provides an on-shell version of the WAdS 3 boundary conditions [59,61] with a positive cosmological constant, in the spirit of the Bañados metrics [126] for AdS 3 (see e.g.[127] and references therein).Appendix C gives extra details on the warped conformal field theory derivation of the entropy.Have fun reading!

Geometry of Warped Flat Spacetimes
In this section we consider the geometry of the warped flat spacetimes mentioned in the introduction.We start by discussing warped flat space in subsection 2.1 and compute its global Killing vectors and the finite coordinate transformations generated by them.Then, we introduce a particular quotient of warped flat space in subsection 2.2.Depending on the parameters of the warped flat quotient, the spacetime may contain closed timelike curves.We discuss the causal structure of both warped flat space and the warped flat quotient in section 2.3 using techniques developed in [128] and find that both spacetimes are, in fact, not black holes.In particular, the causal diagrams of warped flat space and the warped flat quotient in the case where no closed timelike curves are present coincide with the one of Minkowski space.We find that for the case where closed timelike curves are present, the causal diagram for the warped flat quotient is the same as that of a flat space cosmology.

Warped Flat Space
We consider the following three-dimensional spacetime [63,113]: which was dubbed warped flat spacetime in [63].Here, the coordinates τ, x, y range over the real numbers.The metric (2.1) is smooth as far as curvature invariants are concerned, which read R = 6/2 , R µν R µν = 108/ 4 .The Cotton tensor (which is analogous to the Weyl tensor in higher dimensions) reads: and C µν C µν = 2592/ 6 2 The metric (2.1) can be obtained as a limit of global warped Anti-de Sitter space (WAdS) or warped de Sitter space (WdS) (both with positive cosmological constant) [63].Detailed discussions of the appropriate limits are provided in Appendix A. The inverse metric in coordinates (τ, x, y) reads 3) it can be seen that the normal vector to constant τ surfaces is always timelike while the one to constant x surfaces is spacelike.However, for constant y surfaces, 12 2 n µ n µ = 1 − 12x 2 ; so it is a spacelike surface for |x| > 1/ √ 12, while for |x| < 1/ √ 12 it is a timelike surface.The x = ±1/ √ 12 surfaces are null surfaces.
The exact isometries are generated by the four Killing vectors satisfying the following algebra: where I 0 commutes with all other generators.The algebra is precisely the one of the Hamiltonian, annihilation and creation operator of a harmonic oscillator in quantum mechanics, where I 0 is a c-number.The algebra (2.5) is known under the name P c 2 , as it is the 2-dimensional centrally extended Poincaré algebra.We may bring the commutation relations into a wellknown form through the following change of basis Here, H denotes the boost, P 0 and P 1 are the 2-dimensional translations and I 0 denotes the central extension.P c 2 can also be obtained as an İnönü-Wigner contraction of the sl(2, R)+u(1) algebra, as shown in App. A. Finally, it can viewed as a global subalgebra of V ir + û(1) with L 0 = H, P 0 = I 0 , L +1 = a + and P −1 = a − .
We see that some of these isometries have a natural geometric interpretation.
For I 0 and a + + a − = 2∂ τ , the finite coordinate transformations are translations in y and τ .For − 1 2 (a + − a − ) = −τ ∂ y + ∂ x , we have the simultaneous transformation Here, C is an arbitrary constant.Finally the finite transformation generated by H is the most complicated one: We can see that the τ and x transformations are simply boost transformation, while the y transformation is non-trivial and does not allow for a simple geometric interpretation.
The inverse metric (in (u, ρ, ϕ) coordinates) reads (2.12) The global Killing vectors H and I 0 as well as the local Killing vectors a ± are given by ) ) The inverse transformation of (2.9) reads For the parameter range ω ≤ −ξ/2 the metric component g ϕϕ becomes negative and closed timelike curves occur in the region ρ > 0 between ρ 1 = −ω − −ξ(ξ + 2ω) and In the following we will differentiate between the two cases of interest: 1.For ω > −ξ/2: no closed timelike curves appear 2. For ω ≤ −ξ/2: closed timelike curves appear in the region In the case ξ + ω = 0 closed timelike curves start to occur at ρ = 0, which is why we will restrict to ξ + ω > 0 in the following.The surface ρ = 0 is a Killing horizon of the following Killing vector

Causal Structures
In this section we discuss the causal structure of warped flat space (2.1) and the warped flat quotient (2.10), using methods developed in [128].In [128] a new class of two-dimensional diagrams, the so-called projection diagrams, were introduced as a tool to visualize the global structure of spacetimes.These diagrams can be used to depict non-spherically symmetric or non-block diagonal metrics with two-dimensional diagrams, using a two-dimensional auxiliary metric constructed out of the spacetime.For this one uses a map π which maps (a subset of) the spacetime U to (a subset of) R 1,1 .This map is constructed such that every timelike curve in U gets projected to a timelike curve in R 1,1 and each timelike curve in π(U ) is the projection of a timelike curve in the original spacetime.This way causal relations in π(U ) reflect causal relations in U , see [128] for a precise definition.
In regions where closed timelike curves appear, causality is not represented in any useful way in the projection diagram.For this reason these regions are removed from the diagram.

Warped Flat Space
First, we consider warped flat space (2.1).The map π from the definition of the projection diagram is given by the projection (τ, x, y) → (τ, x).The auxiliary metric reads which is two-dimensional Minkowski space R 1,1 , whose conformal compactification and conformal boundaries are well-known.
We now want to answer the question whether the geometry (2.1) possesses a non-zero black hole region.Our notion of future asymptotic infinity I + of (2.16) will be defined with respect to the conformal boundary of the two-dimensional metric.The two-dimensional spacetime (2.16) does not have a black hole region, as the whole spacetime lies in the causal past of This can be seen explicitly by considering the family of null curves (τ (s), x(s)) = (τ 0 + s, x 0 + s) going from each point in spacetime all the way to I + .These curves can be lifted to null curves in the three-dimensional spacetime going through every point: (τ (s), x(s), y(s)) = (τ 0 + s, x 0 + s, y 0 − x 0 s + s 2 2 ).This shows that there is no black hole region in our threedimensional spacetime3 .

Warped Flat Quotient
Next, we consider the warped flat quotient.Here we will differentiate between two cases, the case where CTCs are present and the case without CTCs.As the coordinate system (u, ρ, ϕ) used in the previous section cannot be extended beyond ρ = 0, we will work with the coordinates (τ, x, y) which are everywhere well-defined.We split our spacetime into four sectors: This split is depicted in figure 1.The coordinate transformation in sector I has already been discussed in the previous section (see equation (2.9)).In the sectors II, III, IV we introduce new coordinates u, ρ and ϕ as follows ) Here, we depict the split of the spacetime into the four sectors in the compact coordinates T and X, which will be introduced below.Each coordinate patch u, ρ, ϕ covers one such sector. ) This leads to the metric (2.10) in each sector, which upon identification of (u, ρ, ϕ) ∼ (u, ρ, ϕ + 2π) gives the warped flat quotient.Here, u runs from −∞ to ∞. Depending on the sector ρ runs from −∞ to 0 or from 0 to ∞.The inverse transformation of (2.17) reads ) ) The starting point of the construction is to write the metric (2.10) in the following form: (2. 19) We see that the last term is positive everywhere, except for in the region where closed timelike curves are present (compare with (2.10)).We discuss the case without closed timelike curves first for which the last term is manifestly positive.We project in such a way that the auxiliary metric γ µν reads . (2.20) Then we perform the following coordinate transformation in each of the four sectors ) ) where c = 12ξ ξ+ω and f (ρ) is the solution to the differential equation The solution satisfies that f (ρ = ±∞) = ∞ and f (ρ = 0) = −∞.The coordinates V, U both run from (−∞, ∞).We introduce two more coordinates so that (T − X)/2 ∈ (−π/2, π/2) and (T + X)/2 ∈ (−π/2, π/2).Now we can rewrite γ µν as The projection π is then defined as the map (τ, x, y) → (X(τ, x, y), T (τ, x, y)).The map is differentiable everywhere.The conformal factor reads4      The conformal factor goes to 0 as ρ, u go to ± infinity, is regular at ρ = 0 and is positive everywhere.The projection diagram of the warped flat quotient in the case where no closed timelike curves occur looks like the one of two-dimensional Minkowski space and is depicted in figure 2. The above derivation is valid for the case where no closed timelike curves appear.In the case where closed timelike curves appear, the construction is valid everywhere except for in the region −ω − −ξ(ξ + 2ω) < ρ < −ω + −ξ(ξ + 2ω) which must be excised from the diagram.We thus cut off our spacetime at ρ = −ω − −ξ(ξ + 2ω).The resulting diagram is depicted in figure 3.
We now want to answer the question whether the geometry (2.10) possesses a non-zero black hole region.As before, our notion of future asymptotic infinity I + of (2.16) will be defined with respect to the conformal boundary of the two-dimensional metric.The two-dimensional spacetime (2.20) does not have a black hole region.To show that the threedimensional geometry does not possess a black hole region either we proceed as follows.In each sector we consider null curves in the two-dimensional geometry (2.20).These curves (u(ρ), ρ) are solutions to the differential equation and may be lifted to null curves (u(ρ), ρ, ϕ(ρ)) in the three-dimensional geometry provided that ∂ϕ(ρ) ∂ρ (2.27) The differential equations can be solved to give two curves emanating from every point in spacetime, except at ρ = 0, where the coordinate system breaks down.Considering the curves in the global coordinate system (τ, x, y) (see (2.9) and (2.17)), we find that the coordinates (τ (ρ), x(ρ), y(ρ)) are finite and continuous for any ρ if one patches the curves in the sectors I, II, III, IV together appropriately.As there exist such null curves going through every point this shows there is no black hole region in our three-dimensional spacetime.

Warped Flat Spacetimes in TMG
In this section we will determine the asymptotic symmetries of a consistent phase space including the warped flat quotients.To do so, we will particularize to a specific gravity theory admitting them as solutions.The spacetimes (2.10) solve the equations of motion of Topologically Massive Gravity (TMG) [114] with µ = 3 √ 3 and Λ = 1 2 5 .We will find it convenient to work with the first-order formulation of TMG [123], so that we can use methods developed for general Chern-Simons-like theories of gravity in [124,129].An additional reason for working in the first-order formulation is that it is straightforward to obtain a well-defined variational principle, making it possible to single out a particular symplectic structure and thus fixing the ambiguity of adding boundary terms to the action.In fact, the charges as defined in [124,129] are already compatible with a well-posed variational principle.

First-order Formulation of TMG
The first-order formulation of TMG is given by the action of the form where e a is the dreibein, ω a is the dualized spin-connection and f a is an auxiliary field which enforces the vanishing torsion constraint.The field equations read de a + a bc ω b ∧ e c = T (ω) a = 0 (3.3a) All of the fields appearing in the action (3.2) are Lorentz-vector valued one-forms, hence we can collect them into a single Chern-Simons-like one-form field with an additional (flavor) index a pa = {e a , ω a , f a }.The action (3.2) is then economically expressed as This is known as the Chern-Simons-like (CS-like) action [124,125] and can be used to derive for instance asymptotic charges [130] and asymptotic symmetries [129] for a variety of 3D massive gravity models at the same time.Upon choosing the completely symmetric flavor space metric g pq and structure constants f pqr to be one can easily recover (3.2) from (3.4).
In [129] the formalism to compute charges for asymptotic Killing vectors in CS-like theories was developed.One first specifies certain boundary conditions on the set of fields a p , consistent with the equations of motion (at least, asymptotically to the relevant order in r).The boundary condition contains, besides a specification of what is fixed as one goes towards the boundary, the specification of what is allowed to vary on the boundary and constitutes the state-dependent information of the theory.Then one considers the gauge-like transformations which leave the boundary conditions invariant, up to transformations of state-dependent functions.In general, not all χ p generate gauge symmetries, some of them are related to second class constraints and hence fixed in terms of a p .This is why we referred to them as gauge-like parameters.The gauge-like parameters corresponding to diffeomorphisms take the form χ p = a p µ ζ µ . (3.7) One then uses the obtained χ p to compute the asymptotic charges [124,129] (on the boundary of the disk at a constant time slice) Here the dot denotes contraction with the SO(2, 1) invariant metric η ab .This expression should be integrable and finite in order for the boundary conditions to be consistent.The asymptotic symmetry algebra is then spanned by the Dirac brackets of the charges, which is most easily computed as

Phase Space
While the considerations in section 2 have been coordinate independent, the computation of asymptotic charges relies on the introduction of a coordinate system (u, r, ϕ) and a specification of the fall-off conditions on the metric as r → ∞.In order to compute the asymptotic symmetry algebra of the warped flat quotient (2.10), we find it convenient to make a coordinate transformation to a radial variable r defined as Note that we had assumed ξ > 0 without loss of generality and hence this coordinate transformation keeps us in the future wedge with ρ < 0, where there are never any closed timelike curves.We obtain the metric The advantage of using this metric over (2.10) is that now ξ and ω can be promoted to arbitrary functions of ϕ, while still solving the TMG field equations (3.1) (now with as the coordinate transformation (3.9) changes the spacetime orientation). 6he state-dependent data in this metric are the two functions ξ and ω, which we will now take to be arbitrary functions of ϕ.The next step consists in finding the Chern-Simons-like fields a p consistent with the TMG field equations and leading to the metric (3.10) through g µν = e a µ e b ν η ab .We choose the following Lorentz frame for the dreibein: e r = T 0 , (3.11b) Here T a are SO(2, 1) generators.Solving (3.3a) we find for the components of the spinconnection ) ) (3.12c) Equation (3.3b) can be solved to find the components of the auxiliary field as ) This we will consider to be our boundary conditions for the Chern-Simons-like fields a p .This terminology might be confusing, as these are exact solutions to the TMG field equations and not defined in terms of an asymptotic expansion close to the boundary at r → ∞.It is, however, a common feature in three-dimensional gravity that the asymptotic expansions are finite.For instance, the Fefferman-Graham expansion in AdS 3 terminates at the second order.

Asymptotic symmetry transformations
We will now consider gauge-like transformations (3.6) which leave a p invariant up to transforming the state-dependent functions ξ and ω.These parameters will correspond to asymptotic symmetry transformations with asymptotic Killing vectors obtainable through (3.7).By explicitly solving (3.6), we find that they are given in terms of two arbitrary functions of ϕ, denoted here by T (ϕ) and Y (ϕ), as: where f (ϕ) solves the differential equation This solution is given by for arbitrary constants c and d.These asymptotic gauge transformations preserve (3.11), (3.12) and (3.13) provided that The asymptotic Killing vectors connected with boundary conditions preserving gauge transformations can be calculated via (3.7), and read with f (ϕ) given by (3.16).Note that (3.7) holds only for the dreibein and the auxiliary field.
The equation for ω holds up to a term proportional to ξ ω = f (ϕ) 12rξ T 1 .This term can be removed by a local Lorentz transformation.
Before we compute the asymptotic charges and their symmetry algebra, let us look at the exact Killing vectors for the warped flat quotient (3.10) with constant ξ and ω.The 4 Killing vectors are given by: We observe that the first two correspond to linear combinations of I 0 and H in (2.13a) and are globally well-defined after the identifications.The last two are only globally well-defined and single-valued in the domain 0 ≤ ϕ < 2π when In analogy with AdS 3 gravity, we identify the vacuum as (one of the metrics) with that value of the parameter ξ.

Charge algebra
We now compute the charges from the general formula for Chern-Simons-like theories (3.8).Using , the variation of the boundary charges (3.8) is integrable and finite, but it only depends on the gauge parameters T and Y ; the dependence on f (ϕ) drops out completely } we compute the Dirac bracket algebra of the Fourier modes of the charges, defined as [P n , P m ] = 0 .This is a warped conformal symmetry algebra, but with vanishing central charge and vanishing û(1) level.The asymptotic symmetry algebra is an infinite-dimensional lift of the charges associated to the global Killing vectors I 0 and H, whereas the asymptotic Killing vectors associated to the function f (ϕ) in (3.16) do not have any corresponding asymptotic charges and neither appear in the transformation rules of ξ and ω.Hence these asymptotic Killing vectors do not play any role in the asymptotic symmetry algebra.

Thermodynamics
For constant ξ(ϕ) = ξ and ω(ϕ) = ω the mass and angular momentum read The mass M and the angular momentum L 0 are positive and bounded from below due to our assumptions ξ > 0 and ξ + ω > 0.
The angular velocity and the temperature read The entropy is given as [76,[131][132][133][134]] where7 Here, the functions are defined by the decomposition of the metric in the following form which explicitly reads (3.30d) We obtain the entropy which then satisfies a first law of the following form The sign of the first law is reminiscent of the first law at the inner horizon of black holes [134] or for cosmological horizons [75].We see that the entropy (3.31) is manifestly positive as ξ + ω > 0 and ξ > 0. When expressed in terms of the mass P 0 and angular momentum L 0 the entropy reads The goal in the next section will be to reproduce (3.33) using the warped conformal symmetries (3.22) with vanishing central charges.

Warped Conformal Field Theories at vanishing level
The algebra (3.22) describes a warped conformal field theory with no central extensions.In this section we reproduce (3.33) by using a Cardy-like entropy formula for warped conformal field theories of the form (3.22).This formula can be derived by a slight generalization of the derivation provided in [73], see appendix C for the details.For this we will consider a warped conformal field theory on the torus in Lorentzian signature described by coordinates (t, ϕ) chosen such that the symmetries are The identifications of the coordinates read (t, ϕ) ∼ (t, ϕ + 2π) ∼ (t + iβ, ϕ + θ) .(4.2) These transformations (4.1) are generated by infinitesimal vector fields l n = ie inϕ ∂ ϕ and The algebra of the charges L n , P n on the torus is given by (4. 3) up to central extensions.The algebra (4.3) admits three non-trivial central extensions (c, κ, k) [P m , P n ] = k 2 mδ n+m,0 .
The partition function at inverse temperature β and angular potential θ reads Z(β, θ) = Tr exp (−βP 0 + iθL 0 ) .(4.5) Under a modular transformation the partition function transforms as The modular properties of this partition function were discussed in [73] for warped conformal field theories with (c, 0, k).For L 0 bounded from below we find in the regime of small imaginary θ (provided that the vacuum is gapped) that Using the thermodynamic relation S = (1 − β∂ β − θ∂ θ ) ln(Z) we find on the one hand for WCFTs with (c, κ, k), provided that k = 0, that while on the other hand for warped conformal field theories with generic (c, κ, 0) as advertised in (1.1).Thus, in our case of (c, κ, k) being (0, 0, 0), we obtain (4.8).At this point, we need to identify the vacuum charges.We saw in section 3.3 that the family of metrics (3.10) generically exhibits two globally defined Killing vectors, except at the special value (3.20)where a symmetry enhancement occurs and which we thereby identify with the vacuum value.Using (3.23) this leads to However, this condition is not sufficient to specify the vacuum value of ω.It is known that in Warped CFTs with non-vanishing U(1) level, the vacuum charges are given by [73] For a WCFT dual to WAdS 3 spacetimes in TMG, the values of q, c and k are known.We have repeated the asymptotic symmetry analysis for these spacetimes with positive cosmological constant in appendix B, of which the warped flat solutions can be obtained by the limit ν 2 → 3. The result is We will assume that this relation holds for holographic WCFTs dual to TMG, i.e. a gravity theory where the diffeomorphism anomaly is captured by a gravitational Chern-Simons term with coupling constant ν.In particular, we assume there is a smooth limit of (4.10) as ν 2 → 3 (even though both c and 1/k blow up in that limit) so that at ν 2 = 3: Plugging (4.12) and (4.9) in (4.8), we reproduce exactly the entropy (3.33).

Discussion
We have found a consistent set of boundary conditions accommodating the family of solutions (2.10).Their ASA was shown to consist in a centerless WCFT algebra (3.22).The corresponding field theory has a degeneracy of states given by (4.8) which matches the Bekenstein-Hawking entropy (3.33) of the corresponding solutions.We conclude with some comments: 1.The gravitational theory we have considered is TMG, which we have expressed in first-order formulation.This allowed us to use an unambiguous definition of the charges inherent to Chern-Simons-like theories.It would be interesting to reproduce our results using Iyer-Wald [135] or  charges for TMG in metric formulation.Boundary terms needed for a well-defined variational principle for our boundary conditions might be required to be make the asymptotic charges well-defined8 .Extensions to other higher-curvature theories/theories with matter could also be performed.
2. The entropy formula (3.22) we derived for a centerless WCFT shares interesting similarities with that of a BMS 3 algebra.Indeed, let us perform the (non-local) change of coordinate u = P 0 ũ and let P0 := Q ∂ ũ .It follows that δ P0 = P 0 δP 0 , and hence P0 ∼ P 2 0 .This means that in terms of this "quadratic charge", the entropy reads as . This has the same form as the BMS-Cardy entropy formula (see third column of Table 1).This is reminiscent of the quadratic ensemble in WCFTs, in which the degeneracy of states takes the form of a Cardy formula even in the absence of conformal symmetry [73].In this case, it is not clear whether a BMS symmetry is present or not.It would be interesting to perform an asymptotic symmetry analysis in this ensemble.
3. The matching of the field theory and gravitational entropies required to identify the vacuum charges.We gave a geometric and heuristic argument for these values.However, from a purely field-theoretical point of view, these values should have been vanishing, because of the vanishing of all central extensions (see C.38a).The determination of the vacuum charges on the cylinder using a warped conformal transformation assumes a state-operator correspondance that might not be true in this case.This adds to the fact that highest-weight representations are generically non-unitary for WCFTs with vanishing k, which does not exclude the existence of unitary induced representations.Understanding these aspects of WCFTs is certainly worth exploring.
4. WF spacetime has a 4-parameter family of isometries.When quotienting out the global spacetime by a one-parameter discrete subgroup of the isometry group generated by a Killing vector Ξ, the resulting geometry will generically have as Killing vectors all original Killing vectors commuting with Ξ.This means that, in order to get a geometry with two Killing vectors, one could quotient out WF by a vector of the form αX + βI 0 where X is any Killing vector.The spacetimes studied here correspond to X = H and non-vanishing α and β, while self-dual warped flat is obtained for α = 0 [116].It might be interesting to investigate whether more general quotients could lead to physically meaningful solutions. with: At first sight this metric may look much more inconvenient than (A.5), but the advantage we gained is that we can now replace L and M by arbitrary functions of ϕ and it will still be an exact solution of the TMG field equations (3.1).From now onward we will suppose that M and L are, indeed, arbitrary function of ϕ.The asymptotic Killing vectors which preserve the form of the metric (B.3) as ρ → ∞ are Following the analysis of section 3, we will compute the charges and asymptotic symmetry algebra in the first-order formulation of TMG.We parameterize the metrics (B.3) by a frame field where we are assuming that ν 2 > 3 (similar expressions exist for when ν 2 < 3).The field equations for TMG in first-order form (3.3) can be used to solve for the spinconnection ω and the auxiliary field f .The gauge-like transformations (3.6) which preserve the form of the Chern-Simons-like fields a p = {e, ω, f } are given by (3.7) with ζ given by the asymptotic Killing vectors (B.5), up to a subleading term in the ω-component of χ p .The asymptotic charges (3.8) are integrable and finite in the limit ρ → ∞.They are given by where the state-dependent functions M and L transform under the asymptotic symmetry transformations generated by (B.5) as consistent with the values obtained in [73], upon analytically continuing ν → iν and → i .
The exact Killing vectors of the metric (A.5) become globally well-defined if In the limit to warped flat ν 2 → 3, this translates to the vacuum value of P 0 For this value of P vac 0 , the vacuum value of L 0 is well-defined in the limit ν 2 → 3, even though both c and 1/k diverge:

C Derivations of Warped Cardy Entropy formula
This appendix is meant to provide a reference for Cardy formulas and modular properties of any given WCFT.We first consider the theory to be defined on a complex plane described by coordinates z, w, which will be treated as independent complex coordinates.On this plane T (z) denotes the right-moving energy momentum tensor and P (z) denotes a right moving û(1) Kac-Moody current, which generate coordinate transformations of the form These transformations are generated by infinitesimal vector fields The algebra of charges on the plane where δ is a constant tilt parameter.
For now, we consider a WCFT on the cylinder with coordinates (t, ϕ) chosen such that the symmetries are ϕ = f In the following, we drop the primes for L 0 and P 0 , as the spectrum of the primed operators coincides with the original spectrum [73] .For L 0 bounded from below we find that, provided that the vacuum is gapped, in the regime of small imaginary θ Z(β, θ) ∼ exp n , L ± m ] = (n − m)L ± n+m + c ± 12 (n 3 − n)δn+m,0 [Ln, Lm] = (n − m)Ln+m + c 12 (n 3 − n)δn+m,0 [Ln, Lm] = (n − m)Ln+m + cL 12 (n 3 − n)δn+m,0 [Ln, Pm] = −mPn+m [Ln, Mm] = (n − m)Mn+m + cM 12 (n 3 − n)δn+m,0 [Pn, Pm] = k 2 nδn+m [Mn, Mm] = 0 Global L ± 0,±1 ∼ sl(2, R) ⊕ sl(2, R) ⊂ Vir ⊕ Vir L ± 0,±1 , P0 ∼ sl(2, R) ⊕ u(1) ⊂ Vir + û(1) SWBH = SWCFT SFSC = SBMS 3 a 7 p b J f 8 W c g y y T I S R l y 1 L q l r 0 5 P s y z hC p m k 1 r Y D P 8 V w T A 0 K J v m k 2 M k s T y k b 0 j 5 v O 6 p o w m 0 4 n l 0 7 I a d O 6 Z F Y G 1 c K y U z 9 P T G m i b W j J H K d C c W B X f S m 4 n 9 e O 8 P 4 K h w L l W b I F Z s v i j N J U J P p 6 6 Q n D G c o R 4 5 Q Z o S 7 l b A B N Z S h C 6 j o Q g g W X 1 4 m j f N K 4 F e C + 4 t y 9 S a P o w D H c A J n E M A l V O E O a l A H B o / w D K / w 5 m n v x X v 3 P u a t K 1 4 + c w R / 4 H 3 + A B P R j s k = < / l a t e x i t > ⇢ < 0 < l a t e x i t s h a 1 _ b a s e 6 4 = " l i X 5 u U A D X 1 t l 4 C u f E c H R w 5 7 K C z U = " > A A A B 7 X i c b V A 9 S w N B E J 3 z M 8 a v q K X N Y h Cs w p 0 I W l g E b S w j m A 9 I j r C 3 2 U v W 7 O 0 e u 3 N C C P k P N h a K 2 P p / 7 P w 3 b p I r N P H B w O

r x 3 7 2
P e u u L l M 0 f w B 9 7 n D x D H j s c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l i X 5u U A D X 1 t l 4 C u f E c H R w 5 7 K C z U = " > A A A B 7 X i c b V A 9 S w N B E J 3 z M 8 a v q K X N Y h Cs w p 0 I W l g E b S w j m A 9 I j r C 3 2 U v W 7 O 0 e u 3 N C C P k P N h a K 2 P p / 7 P w 3 b p I r N P H B w O

r x 3 7 2
P e u u L l M 0 f w B 9 7 n D x D H j s c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l i X 5u U A D X 1 t l 4 C u f E c H R w 5 7 K C z U = " > A A A B 7 X i c b V A 9 S w N B E J 3 z M 8 a v q K X N Y h Cs w p 0 I W l g E b S w j m A 9 I j r C 3 2 U v W 7 O 0 e u 3 N C C P k P N h a K 2 P p / 7 P w 3 b p I r N P H B w O

r x 3 7 2
P e u u L l M 0 f w B 9 7 n D x D H j s c = < / l a t e x i t > < l a t e x i t s h a 1 _ b a s e 6 4 = " l i X 5u U A D X 1 t l 4 C u f E c H R w 5 7 K C z U = " > A A A B 7 X i c b V A 9 S w N B E J 3 z M 8 a v q K X N Y h Cs w p 0 I W l g E b S w j m A 9 I j r C 3 2 U v W 7 O 0 e u 3 N C C P k P N h a K 2 P p / 7 P w 3 b p I r N P H B w O

r x 3 7 2 Figure 2 .
Figure 2. Projection diagram of the warped flat quotient in the case where no closed timelike curves are present (ω > −ξ/2).The upper and lower sectors correspond to ρ < 0, while the left and right sectors correspond to ρ > 0. The blue lines that divide these sectors are the ρ = 0 lines.The four vertices correspond to ρ → ±∞, u = const.

Figure 3 .
Figure 3. Projection diagram of the warped flat quotient in the case where closed timelike curves are present ω ≤ −ξ/2.The closed timelike curves appear in the region −ω − −ξ(ξ + 2ω) < ρ < −ω + −ξ(ξ + 2ω), which is depicted in red in the left picture.In the right picture we have cut off the spacetime at the place where the closed timelike curves appear.The vertical, red wiggly line is the singularity.

Table 1 .
Three scenarios in 3d gravity