Warped Black Holes in Lower-Spin Gravity

We provide a simple holographic description for a warped conformal field theory (WCFT) at finite temperature. To this end we study the counterpart of warped anti-de Sitter black holes in three dimensions using a lower-spin $\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{u}(1)$ Chern-Simons theory proposed by Hofman and Rollier. We determine the asymptotic symmetries, thermal entropy and holographic entanglement entropy and show that all these quantities are in perfect agreement with the expectations from the dual WCFT perspective. In addition we provide a metric interpretation of our results which naturally fits with our analysis in the Chern-Simons formulation.


Introduction
For the largest part of their existences, Anti-de Sitter (AdS) spaces and Conformal Field Theories (CFTs) have followed fairly independent and lonesome paths. Anti-de Sitter spaces have been around since almost the advent of General Relativity [1,2]. The origins of CFTs can be traced back to statistical mechanics, where they were identified as describing critical phenomena. Later, their importance was recognized in string theory, around at the same time the modern study of two-dimensional CFTs was initiated [3]. Signs of interactions (apart from general isometry groups considerations) have appeared in 1986 in the work of Brown and Henneaux [4] demonstrating that the asymptotic symmetries of pure AdS 3 gravity consisted in the two-dimensional conformal algebra. This important step establishing a link between AdS and CFT led to unexpected breakthroughs in gravitational physics such as, if one had to name only one, the beautiful interplay between 2d CFTs and black hole entropy, which crystallizes in the famous Cardy formula [5][6][7][8], and eventually culminating in the AdS/CFT correspondence [9][10][11].
Over the last years, an important collective effort has been devoted in extending the principles of AdS/CFT to more general set-ups -most notably to non-AdS backgrounds on the gravity side. This is because the holographic nature of gravity in general, and the Bekenstein-Hawking area law in particular are not supposed to be contingent to AdS spaces (see e.g. [12]). A notorious proposal in the context is the Kerr/CFT correspondence [13], suggesting a holographic duality involving (the near-horizon region of) extremal 4d Kerr black holes (see also [14] for near-extremal BHs). The relevant geometry is the so-called Bardeen-Horowitz metric (or NHEK geometry) with SL(2, R)×U(1) isometry [15], which turns out to be universal [16], but importantly does not include the generic AdS 3 factor familiar from the near-horizon geometry of supersymmetric black holes and allowing to apply AdS/CFT techniques. At fixed polar angle, the NHEK geometry reduces to a three-dimensional metric called (self-dual space-like) Warped AdS 3 (WAdS 3 ). Geometrically, they can be understood as a Kerr-Schild-like deformation of AdS 3 using a chiral Killing vector of its SO(2,2) isometry group (see e.g. [17][18][19]. Depending on the type of the latter, one obtains time-like, space-like and null warped AdS 3 spaces. The former can be identified with Gödel space [20], while performing identifications in the latter two yields a variety of black hole solutions [21][22][23] sharing similarities with the Bañados-Teitelboim-Zanelli (BTZ) black holes [24,25] of AdS 3 gravity, and reducing to them when the deformation goes to zero. The asymptotic behaviour of WAdS 3 spaces differs drastically from that of AdS 3 (in particular, they do not satisfy Brown-Henneaux boundary conditions), so WAdS 3 black holes are often viewed as a prototype for non-AdS black hole holography, possibly allowing to get insights into (the near-horizon geometry of) their higher-dimensional cousin.
The departure from a usual AdS/CFT scenario for WAdS 3 spaces is crucially reflected in their asymptotic symmetries, consisting in the semi-direct product of a Virasoro algebra and an affineû(1) algebra instead of the full two-dimensional conformal algebra [26] 1 . This observation has been taken as a starting point for the study of holographic properties of WAdS 3 , in particular through that of two-dimensional field theories with the corresponding symmetries. This led to the definition of a new type of field theories: Warped Conformal Field Theories (WCFTs). These are two-dimensional field theories breaking Lorentz symmetry and possessing an infinite number of conserved charges satisfying a Virasoro-Kac-Moodyû(1) algebra. Hofman and Strominger [28] showed that under certain generic assumptions, a 2d field theory with translation and chiral scale invariance is either a CFT or a WCFT (this is a warped version of the result of Polchinski that scale invariance implies conformal invariance [29]). It is interesting to observe that, contrary to what happened in the unfolding of AdS/CFT (where both sides had been independently known but did not talk to each other), here gravity suggested the existence of a new class of integrable field theories that had not been encountered so far, of which the mere existence was not even guaranteed a priori. Since then however, many steps have been taken towards defining and analyzing the properties of WCFTs: derive a Cardy-type formula and matching with black hole entropy [30], finding explicit examples and calculating partition functions [31], studying phase transitions [32] as well as correlation functions [33], matching of one-loop determinants [34], calculating entanglement entropy [35,36] and the study of anomalies [37].
Facing this new challenge, a natural approach would be to set up the simplest holographic toy-model model capturing the properties of a WCFT and see how far one can get. What we are looking for is the warped counterpart of pure Einstein-Hilbert gravity with a negative cosmological constant for three-dimensional gravity and AdS/CFT. Thanks to the absence of degrees of freedom and the presence of BTZ black holes in its spectrum, pure 3d gravity might be the best candidate for a solvable model with quantum black holes. The latter theory has been shown to exhibit an extremely rich structure [38,39]. WAdS 3 spaces, on the other hand, are not Einstein spaces. For that reason, the holographic models considered so far for WCFTs either consist of higher-curvature gravity theories (see e.g. [26,[40][41][42][43][44][45][46][47][48][49]) or require couplings to matter(see e.g. [19,27,[50][51][52][53][54][55][56][57][58][59]). In any of the cases, the models possess local degrees of freedom. A noticeable exception is the model, dubbed lower-spin gravity proposed in [60], that can be described using a SL(2, R)×U(1) Chern-Simons theory. It was argued that this model is the minimal setup for the holographic description of WCFTs, much like the SL(2, R)×SL(2, R) Chern-Simons theory, which is classically equivalent to the Einstein-Hilbert action [61], is the minimal setup for 2d CFTs. In this paper, we will be concerned with the study of the thermodynamic properties of this model.
In AdS 3 /CFT 2 , one can essentially indistinctly work either in the metric or in the Chern-Simons formulation, as long as one is concerned with semi-classical considerations [38].
Most of the quantities on one side can almost unambiguously be defined on the other side. For instance, determining the Hawking temperature can be done be requiring the absence of a conical singularity in the Euclidean metric close to the horizon, which in the Chern-Simons formulation amounts to requiring that the holonomy of the connection along a certain cycle is trivial. We will identify the gauge connection counterparts of WAdS 3 black holes, study their thermodynamics and compare to the predictions from WCFT.
Another motivation to study warped black hole solutions in lower-spin gravity is related to its similarity to higher-spin gravity theories in AdS 3 that can also be described in terms of a Chern-Simons connection with a specific gauge algebra (see e.g. [62][63][64][65]). For the higher-spin theories in AdS 3 this algebra is sl(N, R) ⊕ sl(N, R) [62]. Since things such as an event horizon are not gauge invariant objects any more, as soon as higher-spin symmetries are present, one needs to find other ways to define thermodynamically sensible black holes with higher-spin charges. In the Chern-Simons formulation this is usually done by requiring that the holonomies of the gauge connection satisfy certain requirements (see e.g. [66,67]). These requirements are basically that the holonomies of the higher-spin connections have the same eigenvalues as the corresponding connection describing the BTZ black hole in AdS 3 . Thus if one is interested in possible higher-spin extensions of warped black holes in a Chern-Simons formulation, one first needs to understand how to describe an ordinary warped black hole in this setup. Providing basic understanding of warped AdS 3 black holes in terms of a lower-spin Chern-Simons theory is another motivation for this work.
In order to describe the thermodynamics of space-like warped AdS black holes in the Chern-Simons formulation, we initially need to make three assumptions that are inspired by similar conditions appearing in the context of the thermodynamics of the BTZ black hole in AdS 3 [24,25] 2 : • Mass and angular momentum are the canonical boundary charges that are associated to translations in a time-like and angular direction respectively. Thus, we assume that the following relations between the Chern-Simons gauge parameters, ε for sl(2, R) andε for u(1), and associated Killing vectors ξ µ holds: where A and C respectively are the sl(2, R) and u(1) Chern-Simons gauge fields.
• The inverse temperature β and angular potential Ω as functions of mass and angular momentum are determined by assuming that the radially independent parts of the Chern-Simons connections A and C, denoted by a ϕ , a t , c ϕ and c t , satisfy the following γ is an undetermined parameter (to be determined later) and L 0 and S respectively are generators of sl(2, R) and u(1). Thus, in order to have sensible thermodynamics, we are assuming that the eigenvalues of h coincide with the eigenvalues of 2πL 0 and similarly the eigenvalues ofh coincide with the eigenvalues of 2πγS.
• We assume that the vacuum solution of the space-like warped AdS 3 black hole in this Chern-Simons formulation is defined by having a "warped-trivial" holonomy around the ϕ-cycle: At this point, it may not be completely clear how these assumptions are justified, but we will take great care in the main body of the paper to show the validity of these assumptions.
This paper is organized as follows. In Section 2 we define our setup, calculate the asymptotic symmetry algebra and the thermal entropy in the Chern-Simons formulation. In Section 3 we show how to define the vacuum state of our configuration in the Chern-Simons formulation and give supporting arguments for the assumption (1.2) using WCFT arguments as well as (1.4). Section 4 will be concerned with computing holographic entanglement entropy using Wilson lines. This computation also provides an independent check of the thermal entropy and thus in turn also the validity of the conditions (1.2). Section 5 will be focused on a metric interpretation of the Chern-Simons results. This metric interpretation provides another explanation for the validity of all our assumptions and, in addition, allows us to fix the previously undetermined parameter γ in terms of geometric variables. Finally, the conclusion and outlook of this work are summarized in Section 6. For comparison, in Appendix A, we summarized how the usual BTZ black holes are described in the metric formulation as well as in the Chern-Simons formulation with sl(2, R) ⊕ sl(2, R) gauge symmetry.

Space-like WAdS 3 Black Holes in a Chern-Simons Formulation
In order to describe space-like warped AdS 3 black holes, we use a sl(2, R) ⊕ u(1) Chern-Simons formulation, very similar to the one presented in [60]. In accordance with [60], we also call it lower-spin gravity. The action is given as follows: where . . . is an appropriate invariant bilinear form, k is the Chern-Simons coupling, κ the u(1) coupling and M a 2+1-dimensional manifold. The gauge field A takes values in sl(2, R) and the second gauge field C in u(1). We take the topology of the manifold M to be a cylinder with coordinates 0 ≤ ρ < ∞, −∞ < t < ∞ and ϕ ∼ ϕ + 2π. Here t is a temporal coordinate while ρ and ϕ are spatial. Choosing the basis 4 of sl(2, R) (L n ) and Some comments are in order regarding the way we present the gauge algebra here, especially the u(1)-part. Introducing explicitly the generator S and the corresponding invariant bilinear form (2.3) might seem unnecessary at first sight. However, we stress that one of our motivations for this work is to provide a reference example of space-like warped AdS 3 black holes in the Chern-Simons formulation that can be possibly extended to include higher-spin excitations. The above choice is useful for this purpose. In the well-known AdS 3 higher-spin case, the embedding of the pure gravity sector into the higher-spin sector determines what kind of higher-spin fields are present in the resulting higher-spin theory (see e.g. [63]). In a similar spirit we suggest that the field content of possible higher-spin extensions of space-like warped AdS 3 black holes is determined by how the basic symmetries (2.2) are embedded into the higher-spin symmetries, sl(2, R) ⊕ u(1) → sl(N, R). 4 A matrix representation of the sl(2, R) part is given by

Boundary Conditions and Asymptotic Symmetries
After the introduction of the basic setup for describing space-like warped AdS 3 black holes in this work, the next step is to write down boundary conditions that include such black hole solutions. Since it has already been shown in [60] that a model like (2.1) can describe space-like, light-like and null warped AdS 3 , we will take the boundary conditions presented in [60] as an inspiration to write down suitable boundary conditions for a spacelike warped AdS 3 black hole 5 .
First we will use some of the gauge freedom to fix the radial dependence of the gauge fields A and C as From the Chern-Simons perspective, the exact form of the group elements b does not have any relevance for computing asymptotic symmetries as well as thermal properties of the physical system described by the Chern-Simons theory. The exact form of this group element, however, is important for the geometrical interpretation of the boundary conditions presented in this subsection. Since we will also present a metric interpretation in Section 5, we fix the group element to be b(ρ) = e ρL 0 . (2.6) There are in fact two reasons to choose this specific expression. The first reason is that this is a very common choice 6 for AdS 3 gravity in the Chern-Simons formulation since the resulting metric takes the form of a Fefferman-Graham expansion. The second one is that this kind of gauge has also been used in [60] and thus is helpful when making contact with the results in this work.
Using this gauge, we propose the following boundary conditions: 7c) 5 Later on we will argue that depending on the choice of chemical potentials the boundary conditions of [60] even include space-like warped AdS 3 black holes. 6 There are also other possible choices such as in e.g. [68].
Here L := 2π k L − 2π κ K 2 and the functions L, K, µ and ν are in principle arbitrary functions of t and ϕ. With a bit of hindsight we interpret the functions L and K as functions characterizing the physical state and the functions µ and ν as chemical potentials 7 . This means in particular that we assume those chemical potentials to be fixed, i.e. δµ = δν = 0. The functions ω a are fixed by the equations of motion, i.e. dA + [A, A] = 0 and dC = 0. For arbitrary but fixed chemical potentials, the equations of motion determine the time evolution of the state-dependent functions as well as ω a as where prime denotes a derivative with respect to ϕ.
The next step is to find the gauge transformations that preserve the boundary conditions (2.7). They are given by These gauge transformations lead to the following infinitesimal transformation behavior of the functions L and K: In addition, the gauge parameters have to satisfy Accordingly the variation of the canonical boundary charge is given by 8 which can be directly integrated to obtain the canonical boundary charge Using this canonical boundary charge as well as (2.12), one can readily determine the following Dirac bracket algebra: where all functions appearing on the r.h.s are functions ofφ and prime denotes differentiation with respect to the corresponding argument. We have also defined δ ≡ δ(ϕ −φ) and δ ≡ ∂ ϕ δ(ϕ −φ).
One can also expand the functions L, K and delta functions in terms of Fourier modes as and then replace the Dirac brackets with commutators using i{·, ·} → [·, ·]. In the end, we obtain the following commutation relations: with c = 6k. This algebra is a semidirect sum of a Virasoro algebra and an affineû(1) current algebra, matching with the basic symmetry for WCFTs. This also coincides with the asymptotic symmetry algebra found in [60]. This is not a surprise, since the ϕ-part of the connections A and C coincide with that of the corresponding gauge fields in [60].

Thermal Entropy: Mass, Angular Momentum and Holonomies
We now proceed in showing that the boundary conditions (2.7) do contain space-like warped AdS 3 black holes. In this section we do this by determining the thermal entropy of the configuration (2.7). We note that from now on we will assume that the chemical potentials µ and ν as well as the state-dependent functions L and K are constant. Under this assumption, the connections a t and c t simplify as Starting with this setup, the procedure in determining the thermal entropy of such solutions is roughly as follows: • Identify mass and angular momentum with the charges that generate time and angular translations respectively.
• Impose suitable holonomy conditions to fix the inverse temperature and angular potential as functions of mass, angular momentum.
• Integrate the first law of black hole thermodynamics to obtain the thermal entropy.
Some comments are in order. The first comment is related to determining the mass and angular momentum of the configuration (2.7). In the usual metric formulation of Einstein gravity, the mass and angular momentum are associated with the charges of the Killing vectors ∂ t and ∂ ϕ respectively, and there is a precise way of relating these charges with the ones determined in the Chern-Simons formulation (see e.g. [71][72][73]). This, however, requires some geometric input to be sure that one is identifying the correct quantities as mass and angular momentum in the Chern-Simons formulation. In this section we will assume that a similar relation holds also in the case at hand. We will show later in Section 5 that this assumption is indeed valid.
The second comment is related to the holonomy conditions we are proposing below in this section. At first we will motivate these conditions based on the first law of of black hole thermodynamics. We stress that the exact form of these conditions at this point of the computation is, similar to the way we determine mass and angular momentum, an educated guess in the absence of a geometric interpretation. In Section 5 we will argue that these holonomy conditions are, indeed, a sensible choice by determining the inverse temperature and angular potential using a metric interpretation of the boundary conditions (2.7) and showing that these expression match precisely with the ones obtained from the Chern-Simons description with the proposed holonomy conditions. In addition one can also use WCFT arguments to show that the proposed holonomy conditions yield the expressions for inverse temperature and angular potential expected from a WCFT perspective.
Now we move to determining the mass and angular momentum. In a Chern-Simons description of three-dimensional spacetimes such as AdS 3 , the gauge parameters ε preserving the connection A are related on-shell to the Killing vectors ξ µ of the corresponding spacetime via the relation = ξ µ A µ . Since the description used in this work is similar to the situations encountered in e.g. [71][72][73], a reasonable assumption is that the gauge parameters (2.10) preserving the connection (2.7) are related to the corresponding Killing vectors via ε = ξ µ A µ ,ε = ξ µ C µ . With this assumption one can determine (the variation of) mass and angular momentum of the solutions (2.7) via For the boundary conditions (2.7) one obtains the following expressions Fixing the chemical potentials µ and ν is tantamount to fixing "units" to measure the energy. One possible choice for example is µ = 0 and ν = 1 for which mass and angular momentum are given by For this choice of µ and ν, one exactly recovers the boundary conditions of [60]. Thus in this case the boundary conditions found in [60] also contain space-like warped black hole solutions.
If the connection (2.7) describes a warped AdS 3 black hole, then the thermal entropy also has to satisfy the first law of black hole thermodynamics where S Th is the thermal entropy, β the inverse temperature and Ω the angular potential. We have already determined what (the variation of) mass and angular momentum is in the previous paragraph. The only missing ingredients in order to determine the variation of the thermal entropy δS Th are the functional relations between the inverse temperature β, the angular potential Ω and the mass M and angular momentum J. Once these relations are identified, one can functionally integrate (2.23) to obtain the thermal entropy.
In a Chern-Simons theory these functional relations are usually determined by looking at the holonomy of the connection (see e.g. [66,67,74,75]). Thus, in what follows we will be choosing certain holonomy conditions that will fix β and Ω in terms of the state-dependent functions L and K. Before stating those conditions, it will be illuminating to rewrite the first law (2.23) as β Ω dϕ c ϕ δc ϕ (2.24) or, in a little bit more suggestive manner as One can see already at this stage that if h is proportional to the elements of the center of the relevant gauge algebra then two things happen: • The expression for the variation of the entropy (2.25) can be trivially integrated.
• One makes manifest that all the relevant information regarding entropy is encoded in the connections along the non-contractible cycle that wraps around the horizon. Or, in other words, the thermal entropy can be computed by using a Wilson line wrapping around the horizon [76,77].
Moreover, the holonomies of the rotating BTZ black hole for example also take the form where Eigen[...] denotes a set of eigenvalues for ... and γ is some constant. Imposing these conditions, one finds that the holonomies of h andh are given by Therefore, if γ is an integer for example one finds that the holonomy lies in the center of the gauge group, in close analogy to the BTZ case. However, since our goal is to describe a warped geometry, we do not assume that this has to be necessarily the case. The reason is that warped AdS 3 spacetimes are deformations of AdS 3 spacetimes and, as such, it is suggestive that a similar kind of "deformation" also happens at the level of holonomies.
Enforcing the conditions (2.27) fixes the inverse temperature and angular potential as Both the inverse temperature and the angular potential are theory-independent quantities and only depend on the geometry in question. Thus, we will use the metric interpretation of Section 5 later on to fix the exact value of γ.
After this, one can directly proceed in determining the thermal entropy via functionally integrating (2.25) to obtain For the choice µ = 0 and ν = 1, the thermodynamic potentials are given by and one obtains the following thermal entropy in terms of mass and angular momentum: where c = 6k. This is exactly the form of the entropy for a space-like warped AdS 3 black hole and at the same time that of a WCFT at finite temperature [30]. Since for this choice of chemical potentials the mass and angular momentum are directly related to the zero modes of the functions L and K, we will assume that µ = 0 and ν = 1 in everything that will follow from now on. This expression for the entropy is already a strong evidence supporting that the connection (2.7) correctly describes space-like warped AdS 3 black holes.
Since the holonomy conditions (2.27) at this point of the computations are merely assumptions, we will devote the rest of this paper to show that these conditions are sensible choices for space-like warped AdS 3 black holes.

Vacuum State and WCFT Entropy
The expression for the entropy (2.32) takes the form expected for the thermal entropy of a WCFT. A natural question is then how to relate these two quantities to each other. The general formula for the thermal entropy for a WCFT at finite temperature is given by [30] In this expression M and J are mass and angular momentum respectively and M v and J v are those of the vacuum. Thus, in order to make contact with this formula, one first needs to determine what the vacuum solution of (2.1) is.
In the usual metric formulation of Einstein gravity, one way to determine the vacuum is via the maximal number of globally well-defined Killing vectors. Looking at the BTZ black hole for example one finds in general six linearly-independent Killing vectors 9 . However, out of these six Killing vectors, only two are globally well defined for general values of mass and angular momentum: the ones associated to time and angular translations. One possible definition of a vacuum state is that it is the state with the highest amount of symmetry. Thus, in this case it should be the state where all the six Killing vectors are well defined globally. For the BTZ black hole, this happens for very specific values of the mass and angular momentum (see (A.18)) and yields global AdS 3 .
For space-like warped AdS 3 black holes, the situation is similar; the only difference is that one generically has four Killing vectors out of which two are again globally well-defined for any value of mass and angular momentum. In a Chern-Simons theory the role of Killing vectors is taken by the gauge parameters (2.12) and as such one might expect that one encounters similar features at the level of the gauge parameters. Another way to look for the vacuum in a Chern-Simons formulation is to use the holonomies. Taking again the BTZ black hole as inspiration (see (A.17)), it is straightforward to check that the holonomies around the ϕ-cycle are in general non-trivial. Only for very specific values of mass and angular momentum -the ones that give global AdS 3 -these holonomies become trivial. In the following we will use this as a guiding principle to determine the vacuum state of the solution (2.7).
Inspired by this, we assume that for the vacuum state of the warped AdS 3 case the holonomy along the ϕ-cycle obeys b −1 e aϕ b = −1l, e cϕ = e 2πiγ .
or on terms of mass and angular momentum (again for the choice µ = 0, ν = 1) The integer n accounts for the periodicity of the complex exponential function and leads to additional possible branches of solutions 10 . In the following we will choose the branch with n = 0 in order to make contact with the WCFT vacuum values found in [30].
In order to check if the vacuum values lead to some kind of symmetry enhancement at the level of the gauge parameters, we first determine these gauge parameters satisfying δ ε A = δεC = 0. From these equations, one finds that the gauge generators defined in For µ = 0 and ν = 1, the solutions to these differential equations are given by Since the manifold M has the topology of a solid cylinder whose ϕ coordinate is 2πperiodic, one also has to require that (ϕ) ∼ (ϕ + 2π) as well as σ(ϕ) ∼ σ(ϕ + 2π). However, looking at (3.6) one sees that for generic values of L this is only true if C 1 = C 2 = 0. This fits perfectly into the picture of a space-like warped black hole: out of the four linearly-independent Killing vectors, only two are globally well-defined. On the other hand, if L and K take the values found in (3.4), one sees that √ L = i 2 and thus (3.7b) These gauge parameters are 2π-periodic for any values of the four integration constants C n . This fits perfectly with what we expected at the beginning of this section: there should be four linearly-independent globally well-defined gauge parameters for the vacuum state.
After having determined the vacuum, one can now make contact with the general formula of a WCFT at finite temperature (3.1). Plugging the vacuum values (3.4) into that expression one obtains This is exactly the same form as the entropy derived in the gravitational setup (2.32). We note that (3.4) indicates the parameter γ introduced in (2.27) is related to the vacuum value of the mass.
Having shown that the vacuum values (3.4) in combination with the general WCFT formula (3.1) yields the same expression as the thermal entropy (2.32), we will now present a WCFT argument that supports the choice of holonomy conditions (2.27). Starting from the WCFT formula (3.1) one can determine its variation with respect to the charges J and M . This allows one to identify the inverse temperature β and angular potential Ω completely in terms of vacuum values and charges by using the first law (2.23). This yields the following expressions: Inserting the vacuum values (3.4) one immediately recovers (2.31). This is additional evidence supporting the assumption (2.27).

Holographic Entanglement Entropy and Thermal Entropy from Wilson Lines
In this section we compute the holographic entanglement entropy of our setup by using Wilson lines 11 as described in [35]. With the Wilson line method, we will also give an alternative way of computing the thermal entropy of the configuration (2.7). As we will see, this computation also helps us to understand the physical meaning of the parameter γ. In addition this computation gives an additional non-trivial check that the holonomy conditions (2.27) are, indeed, a sensible choice.

Holographic Entanglement Entropy
The concept that entanglement entropy can be holographically computed by using extremal surfaces in the bulk has been first made precise in [79,80] and then subsequently generalized in [81]. The fact that one can compute entanglement entropy holographically gives a very beautiful and intriguing relation between geometry and quantum information. In order to extend these original ideas of holographic entanglement entropy also to higher-spin theories in 3d it turns out that the natural generalization in the context of Chern-Simons theories is to use Wilson lines. For Chern-Simons theories with sl(N, R) ⊕ sl(N, R) symmetries, the basic idea to use Wilson lines for computing holographic entanglement entropy has been first made precise in [76,77] 12 . For sl(2, R) ⊕ u(1) 11 Wilson lines are in general very versatile objects to consider in gauge theories. Another interesting application can for example be found in [78] where the authors used Wilson lines to probe Lorentzian eternal higher-spin black holes in AdS 3 .
12 For the case of isl(N, R) that is relevant for flat space holography see [82,83]. For the dual CFT computation including spin-3 currents that put the proposal of [76,77] on an even more solid footing see e.g. [84,85] Chern-Simons theories that should be dual to WCFTs, slight modifications of the methods by [76,77] are necessary and have been done in [35]. Since one can find the detailed construction already very well explained in [35], we will in the following only review the main ideas of the construction and focus on the parts that are essential for the purpose of our work.
The main idea of computing holographic entanglement entropy in WCFTs is using Wilson lines. To be more precise, the negative logarithm of the trace of a Wilson line taken in an appropriate representation computes the entanglement entropy of the boundary region bounded by the endpoints of the Wilson line. In the case of WCFTs, this means where C denotes the path of the Wilson line and R is the chosen representation (usually an infinite-dimensional representation). A more physical interpretation of this procedure is based on a massive (and spinning) particle travelling along the bulk: the path C describes the world line of the massive particle. For WCFTs, it was argued in [35] that the appropriate representations for the sl(2, R) part are given by a single particle living on AdS 2 . Following the construction in [35], one can find that the leading order piece for the entanglement entropy of the sl(2, R) part of the Wilson line that is attached to some cutoff surface ρ 0 very close to the boundary is essentially given by with where At this point it is important to note that a WCFT is not a relativistic quantum field theory. As such entanglement entropy will look different for different observers. Thus the Wilson line is attached at the initial point (ρ 0 , ϕ i , t i ) and the final point (ρ 0 , ϕ f , t f ). See Figure 1 for a graphical depiction. For the connection (2.7), under the assumption that the entangling interval ϕ f − ϕ i ≡ ∆ϕ is very large compared to the UV cutoff = 2e −ρ 0 i.e. ∆ϕ 1, one finds that  where and c = 6k. Identifying the quadratic Casimir in terms of the central charge as c 2 = k 2 2 = c 2 72 and taking the semiclassical limit i.e. k → ∞, one can solve for ∆α and thus obtain the following expression for the sl(2, R) part of the entanglement entropy 13 (4.8) 13 Since the cosh is an even function one obtains different signs depending on the sign of ∆α. Here we chose to solve for the branch where the entanglement entropy has a positive overall sign.
The u(1) part of the computation is simpler to perform since the Wilson line in this case is given by the integral along the world line of the massive particle as 14 W u(1) (4.9) Thus one obtains as the full expression for the holographic entanglement entropy (4.10) After using the vacuum expressions for M v found in (3.4), this turns into It is instructive to compare the expression (4.10) with the general expression 15 derived in [36] S In this formula δ is a parameter related to the tilt of the cylinder on which the dual WCFT is defined. One can reproduce our expression (4.10) for entanglement entropy by substituting β (2.29) and β ϕ (4.7) into (4.12) and then using the identification δ = 2πγ. Thus the (seemingly) arbitrary parameter γ encountered before is nothing but the tilting parameter of the cylinder. This is also in good agreement with the vacuum values (3.4), since the origin of the non-zero vacuum value of the mass is precisely the tilt of the WCFT cylinder.
The holographic entanglement entropy of our configuration (2.7) provides us with additional evidence for the validity of the holonomy conditions (2.27). To see this, we notice that the thermal entropy (2.32) can also be recovered from the entanglement entropy at a given constant time slice in the limit where the entanglement entropy becomes extensive (see e.g. [77]). That is, in the limit where ∆ϕ βϕ 1 one has Thus one has a way of determining the thermal entropy without any reference to the chosen holonomy conditions. Performing this extensive limit one obtains (4.14) 14 One might wonder why the prefactor in front of the integral is proportional to the vacuum value of the mass. One way to see this is by arguing that the resulting expression for the entanglement entropy should be in accordance with the general formula (4.12). 15 Please note that there is a sign difference with respect to the time interval ∆T used in [36] and the time interval ∆t used here. The conventions for time used in this work are consistent with the ones used in [35].
Thus the extensive limit of the entanglement entropy (4.11) yields again exactly the thermal entropy found previously (2.32), justifying the validity of the holonomy conditions imposed for the derivation of the thermal entropy in Section 2.2.

Thermal Entropy Using Wilson Lines
One of advantages of using Wilson lines to compute entanglement entropy holographically is that one can also directly compute the thermal entropy of black hole solutions (aside using the extensive limit) by simply wrapping the Wilson line around the black hole horizon. In the case at hand this is the ϕ-cycle. Thus, the initial and final points that appear in the matrix M in (4.4) are now identical and one has to solve Solving for ∆α and plugging the result into (4.2) one obtains for the sl(2, R) part The u(1) part is again simply the world line of the massive particle around the ϕ-cycle and thus yields S u(1) These expressions reproduce exactly the thermal entropy (2.32).
As an addendum it is worthwhile to mention that this prescription of the Wilson line wrapping the horizon is nothing else than the statement that all the relevant information about the thermal entropy is encoded in the connection along the non-contractible cycle (ϕ-cycle in the current case). A convenient shortcut 16 to obtain the thermal entropy is thus by first diagonalizing a ϕ and then use the diagonalized version of the connection denoted by λ ϕ to obtain the thermal entropy as 18) or written in a more suggestive way as

Metric Interpretation
Up until this point, we have done everything based on the Chern-Simons formulation. Even though the matching of the asymptotic symmetries, the number of globally welldefined gauge parameters and the thermal entropy fit the description of a space-like warped black hole, it is hard to see at the level of the solution that the connection (2.7) describes the desired black hole. In addition, there is the parameter γ that we identified with the tilt of the cylinder where the dual WCFT is defined on. From a pure Chern-Simons perspective, this is a free and, up until now, undetermined parameter. However, one would expect that this parameter should be fixed in one way or the other by the geometry that is described by this Chern-Simons theory.
In this section, we will provide a metric interpretation of our solution (2.7) based on [60]. Our motivation is two-fold. The first is to explicitly verify that the connection (2.7) describes a space-like warped AdS 3 black hole. The second is to fix γ. The metric interpretation will also enable us to explicitly compute the Killing vectors as well as the inverse temperature and angular potential, providing more evidence to support the claims made in Section 2.2.

Mapping Connection to Metric
A metric interpretation of the vacuum solutions given by the connection (2.7) (with µ = 0 and ν = 1) has been first worked out in [60]. In the following we will employ the same methods as in [60] to show that the connection (2.7) reproduces the metric of space-like warped black holes. Thus, we will review briefly the main points of [60] that are necessary to translate the connection (2.7) into a metric form.
One of the points established in [60] 17 is a precise relation between the gauge fields A, C in the Chern-Simons formulation (2.1) and geometric variables B that encode the geometry. While the Chern-Simons fields are convenient for many purposes, one needs to determine the geometric variables from these fields in order to understand the geometry described by them. In the current setup, this is done as follows: • Define three linearly independent vectors in sl(2, R), (ζ n 0 , ζ n 1 , ζ n 2 ), and the inverse vectors (ζ 0 n ,ζ 1 n ,ζ 2 n ) satisfyingζ I n ζ n J = δ I J for I, J = 0, 1, 2. Depending on the choice of these vectors, one can obtain either time-like, space-like or null warped AdS 3 for vacuum solutions of (2.7).
• Then using these vectors one can determine the geometrical variables B via The variable c encodes the AdS radius, b encodes the warping parameter and α is related to arbitrary rescalings of the time coordinate t.
• Using the sl(2, R) ⊕ u(1) invariant bilinear form M IJ = ζ n I η nm ζ m J with η nm given in (2.3), one can then determine the metric g µν via Since our claim is that the connection (2.7) describes space-like warped AdS 3 black holes, we choose the following vectors in accordance with [60]: where the notation for the vectors is ζ = (+, 0, −). The metric obtained from the connection (2.7) with µ = 0 and ν = 1 in this way is given by This metric can be brought into a more familiar form by choosing a different radial coordinate r via and denotes the radius of the warped AdS spacetime. If one in addition changes the sign of the angular coordinate ϕ as ϕ → ϕ = −φ and chooses the parameters b, c and α as then one obtains precisely the metric of a rotating space-like warped AdS 3 black hole [22] In order to make the contact with the results derived previously using the Chern-Simons formulation a bit easier, however, we will use the metric (5.4) rather than (5.8) in the rest of this section.

Killing Vectors
The purpose of this section is to show that there are four Killing vectors of the metric (5.4) out of which two are globally well-defined. In addition we will show that the two globally well-defined Killing vectors satisfy ε = ξ µ A µ andε = ξ µ C µ . This justifies our assumption (2.20) for (the variation of) mass and angular momentum.
In order to find the Killing vectors of a spacetime given by a metric g µν , one has to solve £ ξ g µν = 0 for some vector field ξ µ . Solving this equation for the metric (5.4), one obtains, as proclaimed, four linearly independent Killing vector fields: where C n are arbitrary integration constants. We note that the Killing vectors ξ 3 and ξ 4 are well-defined both locally and globally. The Killing vectors ξ 1 and ξ 2 on the other hand are not globally well-defined for generic values of L viz. L and K, because of the periodicity of the angular coordinate ϕ. For the vacuum values (3.4), however, these two Killing vectors become periodic. Thus, the vacuum state has four globally well-defined Killing vectors as expected for warped AdS 3 .
Having now both the explicit expression of the Killing vectors (5.9) as well as the expressions for the gauge parameters (3.6) in the Chern-Simons formulation at hand, we can explicitly verify that we correctly identified mass and angular momentum previously in (2.22). This is simply done by taking the two Killing vectors ξ 3 and ξ 4 . It is straightforward to check that they satisfy ε = ξ µ A µ andε = ξ µ C µ for the gauge parameters given by (2.10) and (3.6).

Thermodynamic Quantities
In this part, we will first determine the location of the horizon for the metric (5.4) and then proceed in showing that the horizon is a Killing horizon. As a next step, we will determine angular potential and surface gravity. In the end, we will show that these expressions exactly match (2.29) found in the Chern-Simons formulation, provided the parameter γ is identified appropriately. This also gives us a way of fixing γ completely.
First one needs to determine the location of the event horizon in the coordinates (5.4).
One particularly simple way of doing this is to find the constant-ρ surface where the determinant of the induced metric changes sign, that is the location where the time-like Killing vector ∂ t changes to be space-like. On constant-ρ slices the determinant of the induced metric γ ij reads This expression is zero for We note that, in the coordinates chosen in (5.4), only ρ + is admissible as the location of the horizon.
The angular potential Ω can then be determined by demanding that the norm of the Killing vector ξ = ∂ t + Ω∂ ϕ that generates the horizon vanishes at the horizon i.e. ξ µ ξ µ ρ=ρ + = 0. This in turn means that the horizon located at (5.11) is a Killing horizon.
Computing the norm of this Killing vector, one finds that the angular potential is given by This is exactly the same expression as (2.31) found previously using the Chern-Simons formulation upon identifying the parameter γ as Since the horizon is a Killing horizon one can also associate a surface gravity and thus temperature to it. The surface gravity κ s can be determined straightforwardly via and for the metric (5.4) one obtains Since temperature T and surface gravity κ s are related via T = κs 2π , one finds for the inverse temperature We note that, if one in addition chooses the Chern-Simons level k as well as the u(1) level κ as 18 the thermal entropy (2.32) coincides with the entropy of space-like warped AdS 3 black holes in topologically massive gravity (see e.g. Section 5 of [30]).

Conclusion and Outlook
In this paper we showed that lower-spin gravity described by a sl(2, R) ⊕ u(1) Chern-Simons theory contains solutions that can be interpreted as space-like warped AdS 3 black holes. We argued that certain holonomy conditions give thermodynamically sensible relations between the canonical charges and the corresponding chemical potentials. The resulting thermal entropy is consistent with the first law of black hole thermodynamics and matches exactly the entropy of a WCFT at finite temperature. In order to support our claim, we also computed holographic entanglement entropy and found again perfect matching with what is expected from a WCFT at finite temperature, a holographic dual of the space-like warped AdS 3 black hole. Furthermore, following the dictionary presented in [60], we provided a metric interpretation of our results. Our results show that a theory of lower-spin gravity provides a simple dual setup for a WCFT at finite temperature.
Our results presented in this paper can be extended in various ways. One possible extension would be to look for space-like warped black hole solutions with additional higher-spin charges. Since our work shows that space-like warped AdS 3 black holes can be described by using a Chern-Simons theory with sl(2, R) ⊕ u(1) gauge symmetry, it is suggestive to look at Chern-Simons theories with extended gauge symmetry which contains sl(2, R) ⊕ u(1) as a subalgebra. The most simple possible example for such an extension might be the non-principal embedding of sl(3, R). Even though this particular case would not be a higher-spin extension in the strict sense (since there are no excitations with spin-s > 2), it would be interesting to look at such a theory as a first step towards more complicated examples. The results found in [86] for null warped AdS 3 in higher-spin gravity suggests that such an endeavor is indeed promising. In this work the authors found boundary conditions for a spin-3 gravity theory in AdS that asymptote to null warped AdS 3 and whose asymptotic symmetries are given by one copy of the Polyakov-Bershadsky W 3 algebra, it seems plausible that W 3 algebra will make its appearance as the asymptotic symmetry algebra of a "higher"-spin extended space-like warped black hole.
Since lower-spin gravity provides a simple dual model for a WCFT, it would be very interesting to consider a supersymmetric extension and then apply the localization techniques used in [91][92][93] to compute the full partition function of lower-spin gravity. This could potentially lead to many new insights regarding WCFTs and quantum gravity in general.
Another interesting direction is related to soft excitations of black hole horizons in three dimensions. Starting with the the work [94], consistent near horizon boundary conditions for BTZ black holes were found that lead to a very simple near horizon symmetry algebra, namely two affineû(1) current algebras. The thermal entropy of the BTZ black hole expressed in these near horizon variables takes a strikingly simple form where J 0 andJ 0 denote the zero modes of the affineû(1) current algebras. Following up on this work there were many checks in different setups [94][95][96][97][98][99][100][101][102] as to how general this result of the entropy is in three-dimensional gravity. Since lower-spin gravity provides a new theory to test the generality of the entropy formula (6.1) it might be interesting to see if one can find consistent near horizon boundary conditions that either confirm or contradict (6.1) in this setup.
formulation of Einstein gravity in Appendix A.1 and then continue in Appendix A.2 to review how these geometric statements translate into the Chern-Simons formulation.

A.1 BTZ Black Holes in the Metric Formulation
The metric of the BTZ black hole [24,25] is given by Here the spatial coordinates r, ϕ take values in 0 ≤ r < ∞, ϕ ∼ ϕ + 2π and the temporal one t is in −∞ < t < ∞. This metric has two horizons r ± , that are given in terms of mass M , angular momentum J and AdS radius as Using a different kind of radial coordinate ρ that is defined via Here we used light-cone coordinates x ± = t ± ϕ for convenience. Having the metric in a form such as (A.6) makes comparison with the Chern-Simons formulation easy and thus we will continue to use this form.
Before proceeding to the Chern-Simons formulation, it is useful to first determine some essential quantities characterizing the black hole in the metric formulation. The first important ingredients are the Killing vectors of the metric (A.6). It is straightforward to verify that the Killing vectors of (A.6) are given by where we used the abbreviations L ≡ 2π k L andL ≡ 2π kL . Because of the periodicity of the angular coordinate ϕ, for generic values of L andL, only two out of these six Killing vectors are globally well-defined i.e. ξ 3 and ξ 6 . The other four Killing vectors ξ 1 , ξ 2 , ξ 4 and ξ 5 become globally well-defined once L andL become negative. Looking at the resulting spacetimes, however, one finds that, for almost all possible combinations of mass and angular momentum satisfying this requirement, the resulting spacetime exhibits pathologies such as closed time-like curves, naked singularities or angular excesses/defects with one important exception. That is, the case with M = − k 2 and J = 0 that yields global AdS 3 . Thus, global AdS 3 can be interpreted as the vacuum state of the BTZ black hole.
In order to determine the entropy of the BTZ black hole one first has to determine the location of the outer horizon in the coordinates used in (A.6). This can be done by either reading the location off from (A.4) or alternatively by looking at the value of ρ where the induced metric of (A.6) on slices of constant ρ vanishes. Both yield the same result, namely Once the position of the horizon is located, it is straightforward to determine the angular potential Ω by requiring that the norm of the Killing vector ξ = ∂ t + Ω∂ ϕ vanishes at the horizon i.e. Subsequently one can also determine the surface gravity κ s of the horizon and thus also the temperature T = κs 2π = β −1 via (A.10) By following this, for the metric (A.6), the explicit form of β and Ω turns out to be The thermal entropy can then be determined via the area law S Th = A 4G and one obtains

A.2 BTZ Black Holes in the Chern-Simons Formulation
Einstein gravity with negative cosmological constant can be described by the difference of two Chern-Simons actions [61,109] 14) The for m, n = 0, ±1 whose invariant bilinear form . . . is give in (2.3). The manifold M is given by a cylinder with coordinates 0 ≤ ρ < ∞, −∞ < t < ∞ and ϕ ∼ ϕ + 2π.
The BTZ black hole [24,25] in this setup is described by the following connection: where a popular choice 22 for b is b = e ρL 0 and L andL are the constants that encode mass M and angular momentum J of the BTZ black hole that we introduced in (A.5). The metric in this formulation is recovered via g µν = 2 2 (A µ −Ā µ )(A ν −Ā ν ) and yields (A.6). The metric in the original BTZ coordinates (A.1) can be recovered by simply replacing ρ with the radial coordinate r according to (A.4).
In the metric formulation all thermodynamic quantities have a clear and geometric interpretation. In the following we show how these geometric statements translate into the Chern-Simons formulation. Now we consider the following questions in the Chern-Simons formulation: • How to identify the vacuum state?
• How to relate inverse temperature β and angular potential Ω with the charges L andL?
• How to determine the thermal entropy?
The main idea to tackle these questions in the Chern-Simons formulation is basically to look at the holonomies of the gauge connections A andĀ. These holonomies around given cycles should be trivial, which is tantamount to requiring that these cycles can be contracted in a smooth manner.

A.2.1 The Vacuum State
Looking at the holonomies around the ϕ-cycle that is where n is some integer number. For n = 0 this corresponds exactly to M = − k 2 and J = 0, i.e. global AdS 3 in terms of a metric interpretation. Thus, these holonomies can serve as a tool to determine the vacuum state.
Alternatively one can also make use of the fact that diffeomorphisms or in other words Killing vectors ξ µ in the metric formulation are on-shell equivalent to gauge transformations of the form δ A µ = ∂ µ + [A µ , ], δ¯ Ā µ = ∂ µ¯ + [Ā µ ,¯ ], (A. 19) with gauge parameters = ξ µ A µ and¯ = ξ µĀ µ . One can show that the Killing vectors (A.7) in combination with the connections (A.16) yields six linearly-independent gauge parameters a and¯ b with a, b = 1, 2, 3 satisfying δ A µ = δ¯ Ā µ = 0. Out of these six gauge parameters, again two are globally well-defined and the other four become globally well-defined only for the vacuum values (A.18). This corresponds to the symmetry enhancement that we have encountered at the level of the Killing vectors in the metric formulation.

A.2.2 Inverse Temperature and Angular Potential
A similar approach based on the holonomies can be taken for computing the thermal entropy of the BTZ black hole in the Chern-Simons formulation. This is usually done by first performing a Wick rotation of the Lorentzian time coordinate t to Euclidean time t E which is then compactified. This changes the topology of the solid cylinder to that of a solid torus. One can introduce complex coordinates (z,z) by analytically continuing the light-cone directions x + → it E + ϕ = z and x − → it E − ϕ = −z. These coordinates are then identified as z z + 2π z + 2πτ,z z + 2π z + 2πτ (A.20) where τ is the modular parameter of the boundary torus. Following e.g. [66,67] (see also [74] or [75]) smooth black hole solutions are defined via the holonomies of the contractible cycles around the constant-ρ torus. This is tantamount to the statement that the holonomies belong to the center of the gauge group [110,111]. For rotating BTZ black holes these holonomies are given by The quantities ω,ω can also be written in terms of the Lorentzian connections a t = a z −az and a ϕ = a z + az (ā t =ā z −āz andā ϕ =ā z +āz). In addition one can use the fact that the modular parameters and the thermodynamic potentials for the BTZ black hole have to satisfy 23 In the absence of rotation Ω = 0, this means that the Euclidean time cycle has to be contractible and thus also the holonomy around that cycle has to be trivial. For the rotating solutions, the contractible cycle is a more general thermal cycle given by (A.22) (or equivalently (A.25)).

A.2.3 Thermal Entropy and First Law
These holonomy conditions give precise relations between the thermodynamic potentials β, Ω and the charges L andL. Once these relations are fixed, there are various ways of computing the thermal entropy from the connections (A.16) (see e.g. [74]). One particular simple way is via integrating the first law of black hole thermodynamics δS Th = β (δM − ΩδJ) . (A. 27) In order to be able to integrate this first law, one needs to know how mass M and angular momentum J are related to the constants L andL. In (A.5) we have already presented this relation without any explanation. We will explain this point here. In the metric formulation mass and angular momentum are the canonical boundary charges associated to translations in time as well as in the angular direction i.e. the Killing vectors ∂ t and −∂ ϕ . The (variation of the) canonical boundary charges in the Chern-Simons formulation is given by where and¯ are gauge parameters that preserve the boundary boundary conditions via δ A µ = ∂ µ + [A µ , ] and δ¯ Ā µ = ∂ µ¯ + [Ā µ ,¯ ]. The link between the charges in the metric formulation and the ones in the Chern-Simons formulation is connected with the on-shell relation = ξ µ A µ ,¯ = ξ µĀ µ between gauge parameters ,¯ and Killing vectors ξ µ . Using this relation one finds that the variation of the mass and angular momentum in the Chern-Simons formulation is given by Once mass, angular momentum as well as the corresponding chemical potentials are properly determined, one can proceed in integrating the first law of thermodynamics (A.27) to obtain the thermal entropy. For the connection (A.16) one obtains exactly the same expressions for β and Ω as in (A.11). This then leads to the thermal entropy (A.12).