On quantization of AdS3 gravity I: semi-classical analysis

In this work we explore ideas in quantizing AdS3 Einstein gravity. We start with the most general solution to the 3d gravity theory which respects Brown-Henneaux boundary conditions. These solutions are specified by two holomorphic functions and satisfy simple superposition rule. These geometries generically have a bifurcate Killing horizon (with a noncompact or not simply connected bifurcation curve) which is not an event horizon. Nonetheless, there are superpositions of these geometries which have event horizon. We propose to view these geometries as “semiclassical fuzzball microstates” of BTZ black holes appearing as superposition of these geometries. The details of quantization of these semiclassical microstates will be discussed in an upcoming work.


Introduction
Three dimensional gravity has been long used as a laboratory for testing quantum gravity ideas [1][2][3][4][5]7]. In three dimensions Einstein gravity is classically equivalent to a Chern-Simons gauge theory and has no propagating degrees of freedom. Nonetheless, it still has non-trivial (quantum and topological) features. In presence of a negative cosmological constant, the AdS 3 gravity, the theory seems to show more interesting features than in the asymptotic flat space, e.g. it admits black hole solutions [9][10][11][12][13]. 1 In this work we will be revisiting the AdS 3 gravity and study some features of its generic solutions.
It is well known that all solutions to AdS 3 gravity, which satisfy R µν = − 2 ℓ 2 g µν , (1.1)
3. neither of x ± are compact.
As we will show in the last case (with noncompact x ± ) there is no obstruction in removing functions f ± and mapping the geometry to AdS 3 in Poincaré patch using the local coordinate transformations. We will therefore only focus on the first two cases with at least one compact direction. We will discuss that these two cases correspond to distinct physical cases and will hence be discussed separately. The case with both x ± compact, as we will show, is correlated with the BTZ black hole "semiclassical microstates". This case will be discussed in sections 2 and 3. When f + (or f − ) is zero, one can consider x + (or x − ) to be compact or not. The case with only x − compact corresponds to geometries whose near horizon limit leads to self-dual AdS 3 orbifold. The class of solutions with f + = 0, which will be dealt with in section 4, are those constructed and analyzed in [21].
The geometries in (1.2) are diffeomorphic to AdS 3 and their large r deviation from the AdS 3 geometry, when both x ± are compact, is given by Brown-Henneaux diffeomorphisms. These diffeomorphisms are specified by the same holomorphic or anti-holomorphic f ± functions. In other words, and as we will show explicitly in the next section, (1.2) are in one-to-one correspondence with Brown-Henneaux diffeomorphisms around BTZ black holes or conic spaces and extend the Brown-Henneaux boundary gravitons to the whole bulk. As discuss in [15], one can associate Virasoro charges to these geometries. For a
To gain a better intuition about the geometries in (1.2), let us consider constant f ± case, when both x ± are periodic. For constant (generic) non-zero and positivef ± one recovers BTZ black hole. It is instructive to rewrite the metric into standard BTZ coordinates. Under the coordinate transformation metric takes the form (1.4) The BTZ inner and outer horizon radii ρ ± andf ± are related as: (1. 5) We note that the coordinate system (1.2) is covering only the region of BTZ geometry which is outside the horizon. As pointed out, Fourier modes of f ± are related to charges of a Virasoro algebra which in turn is the symmetry of a 2d CFT dual to the AdS 3 gravity theory [14,15,22]. In the standard AdS 3 /CFT 2 terminology and for the constant f ± case, thenf ± are related to the energy of the Left and Right sectors of the dual 2d CFT L 0 ,L 0 as [23,24] where c is the Brown-Henneaux central charge [16] of the dual CFT. Note also the positivity condition onf ± . 2 The cases with negative but constant f ± ,f ± ≤ −1/4, correspond to conic spaces, as we will see in the next section. The class of solutions in (1.2) have the following remarkable properties: 1. are in one-to-one correspondence with the Brown-Henneaux diffeomorphisms; 2. for set of solutions with f i ± , any linear combination/superposition of them specified by f ± = i A i f i ± constitute another geometry within the same class.
3. As we will show, for generic non-constant f ± these geometries, while having (bifurcate) Killing horizon, the bifurcation curve is noncompact or not simply connected. Moreover, this Killing horizon is not an event horizon.
Based on these facts, we propose to view the set of geometries with all possible f ± which have the same averagef JHEP07(2014)104 as the "fuzzball" 3 semiclassical microstates corresponding to the BTZ black hole specified byf ± . The mass and angular momentum of the BTZ black hole is given through (1.6), In other words, a BTZ black hole with constantf ± may be viewed as "classical geometries," and geometries in (1.2) with a fixed average of f ± , as "semiclassical" description of BTZ geometries. The last step is to "quantized" geometries in (1.2) to fully identify quantum microstates of a BTZ black hole. In this paper we intend to take the first step and establish the geometries in (1.2) as semiclassical microstates of BTZ black holes. The problem of quantization of these geometries is postponed to the next publication.
The rest of this paper is organized as follows. In sections 2 and 3, we consider the geometries in which both x ± are periodic and in section 4 consider geometries with periodic x − and while x + can be periodic or not. In section 2, we analyze the structure of Killing vectors and isometries of metrics in (1.2). We discuss local and global isometries and discuss the restrictions imposed on Killing vectors due to compactness of x ± . In section 3, we study the horizon structure of geometries in (1.2). We show that these geometries generically have a bifurcate Killing horizon, however, the bifurcation surface (which is a one dimensional curve in the 3d case) is not compact or simply-connected and is not in general an event horizon, except for constant f ± case or when one of f + or f − functions is vanishing. In section 4, we repeat the analysis in sections 2 and 3 for the case when f + = 0 and show that in this case the geometry has a degenerate (extremal) Killing horizon [21], which is also an event horizon if f − (x − ) ≥ 0. These geometries may be viewed as semiclassical fuzzball microstates of extremal BTZ black holes or AdS 3 self-dual orbifold. In section 5, we summarize our results and present the outlook for the second part of this project, quantization of the semiclassical microstates. In three appendices we have gathered some technical details of computations. In the appendix A, we have shown that the algebra of the local Killing vectors is indeed sl(2, R) × sl(2, R). This is done through defining a basic bracket among the functions upon which these Killing vectors fields have been constructed. In the appendix B, we have given brief review of mathematics literature on the solutions to Hill's equation, which appears in our analysis of Killing vectors. In appendix C, we have gathered the analysis making a closer connection between our geometries with f + = 0 and those discussed in [21].

Local and global isometries of the general geometries
To qualify as semiclassical microstates we should study smoothness and isometries and horizon structure of geometries in (1.2). To this end we explore the Killing vectors of this geometry for any given f ± (x ± ). Note that in this section and also in section 3 we assume that f ± are both periodic functions of their argument, and that neither of them is zero (while some of the results in section 2.1 is also true for more general cases, with noncompact x ± or when f ± vanish).

Local isometries, the six Killing vectors
As discussed, these solutions are locally AdS 3 and hence have six Killing vectors. If we denote a given Killing vector by then solving equation L ζ g µν = 0 gives following relations where K ± = K ± (x ± ), prime denotes derivative with respect to the argument and functions K ± satisfy following equations Being two third-order differential equations, the above have 3 × 2 independent solutions. Our arguments below together with discussions in appendix A, establish that, as expected, the geometries are locally invariant under sl(2, R) × sl(2, R). These isometries are also globally defined iff (2.3) have real and periodic solutions. The number of real periodic solutions to (2.3) depends on the form of functions f ± and can at most be six. We remark that h a in (2.2), in the leading order in large r expansion, have exactly the form of Brown-Henneaux diffeomorphisms [16], where in the latter K ± = K ± (x ± ) are two generic functions, and are not subject to (2.3). Therefore, supposing that (2.3) has at least a single real periodic solution for any given f + and similarly for f − , then there is a one-to-one correspondence between Brown-Henneaux diffeomorphisms and the geometries in (1.2). As we will explicitly show, (2.3) has indeed always two real-periodic solutions (one corresponding to K + and one to K − ). As stated in the introduction, our proposal is to identify the geometries with the same average f ± and with only one real periodic K ± as semiclassical microstates of the BTZ black hole.

Constructing the Killing vector generators K a
One can show that solutions to (2.3) are specified through a Schrödinger-type equation with periodic potential If the two linearly independent solutions to the above are denoted as ψ 1 and ψ 2 , then it is easy to verify that provide the three linearly independent solutions to (2.3). N is the normalization constant which is chosen to be

JHEP07(2014)104
(One can readily show that N is a constant of motion for (2.4).) Hereafter, we will choose the normalization for ψ α , α = 1, 2 such that N = 1. As has been shown in the appendix A, based on (2.6) one can define a Poisson bracket structure which explicitly shows that K a 's indeed form an sl(2, R) algebra. In other words, this Poisson bracket structure reproduces Lie bracket of the Killing vector fields given through K a .
To qualify as Killing vectors we should also demand K a 's to be real-valued. Therefore, ψ α are also real-valued. Periodicity of K a 's is then related to periodicity of ψ α . If ψ 1 and ψ 2 are both periodic then all three K a 's are periodic too. However, even if ψ α are not periodic, the Floquet theorem (see appendix B) implies that ψ 1 ψ 2 is always periodic. Therefore, regardless of what f ± are, we always have two compact global U(1) + × U(1) − isometries. This latter is the case associated with BTZ black hole semiclassical microstates.
To gain a better intuition about ψ α and K a , it is instructive to analyze constant f ± cases. One can distinguish two positive or negative f cases: and hence the six local Killings are generated by This case corresponds to a BTZ geometry, where only two K ± 0 are periodic. This is of course compatible with our earlier discussions where the BTZ geometry has two global U(1)'s while is locally sl(2, R) × sl(2, R) invariant. Note that among the other four K's, two are exponentially growing and two and exponentially decaying with x ± .
• For f ± = − 1 4 k 2 ± < 0, 9) and the six local Killings are generated by (2.10) For f ± < 0 case, the Killings are not periodic for generic values of k ± . The k ± = 1 case is special because all six K's are periodic with exactly period 2π. This case with f ± = −1/4 corresponds to global AdS 3 which is the only case with global Sl(2, R) × Sl(2, R) isometry group [5,6].
Integer k ± cases are also special, because K's are periodic for these cases too. One can interpret them as orbifolds of global AdS 3 by Z k ± with the Sl(2, R)/Z k + ×Sl(2, R)/Z k − global isometry. These geometries correspond to conic spaces and may be attributed to particles on AdS 3 [5,6].

Further analysis of global isometries of the generic geometries
The Schrödinger-type equation (2.4) with a periodic potential is known as Hill's equation.
Hill's equation have been extensively studied in mathematics literature e.g. [39] (see appendix B for a brief review). Let the periodic function f (x), f (x) = f (x + 2π), have an averagef (over [0, 2π] range) and a partf which is averaged to zero: Then (2.4) may be written as wheref plays the role of a periodic potential oscillating around zero, and −f is the energy eigenvalues. Eq. (2.12) has the form of standard Hill's equation (B.1) withf = −λ.
One may examine when all three K a 's given in (2.5) are real-valued and periodic. This happens when both of ψ 1 and ψ 2 are real-valued and periodic. This latter of course depends on the details off andf . This question has been addressed in the mathematics literature on the Hill's equation, which is reviewed in the appendix B. For a givenf Hill's equation has periodic solutions for a discrete set off 's which are completely determined in terms off . (Conversely, one can reconstructf through its spectrumf .) The values off corresponding to periodic real solutions are arranged in two descending discrete sequences denoted byf 0 ,f 1 , · · · andē 1 ,ē 2 , · · · , such that · · ·f 4 ≤f 3 <ē 4 ≤ē 3 <f 2 ≤f 1 <ē 2 ≤ē 1 <f 0 , (2.13) and the largest onef 0 has a positive value bounded as There is some value of n wheref n (n ≥ 1) changes sign and becomes negative. For large values of k and smooth functions f ± ,f k andē k respectively approach to −k 2 and −(k−1/2) 2 (with O(1/k 2 ) corrections, cf. (B.10)). This is remarkable, recalling the constant, negative f ± case discussed in the previous subsection: for large k, regardless of the details of f ± , we will have a geometry whose Killing vectors approach the conic space. In other words, for a givenf andf taking negative large values, the geometries in (1.2) may be viewed as "semiclassical bound state" of conic space (particles on AdS 3 ) and Brown-Henneaux boundary gravitons. 4 For given geometries specified through f ± (x ± ), however,f ± = 1 2π 2π 0 f ± does not generically coincide with anyf k 's associated with functionsf ± (x ± ). So, we do not generically have three periodic K a 's. In this case,f may fall into stability or into instability

JHEP07(2014)104
intervals. Iff is in the stability intervals of Hill's equation, solutions ψ's are not periodic, nonetheless they fluctuate/oscillate around a periodic solution. These geometries may hence be viewed as "excitations" around AdS 3 particles (conic spaces). On the other hand iff falls into instability intervals, in particular iff >f 0 , one of the solutions has exponential growth while the other is exponentially suppressed and we have a geometry which may be viewed as "excitations" around a BTZ black hole. In fact these are the geometries which we would like to identify as "semiclassical microstates" of the BTZ black hole.

Horizon structure of the general geometries
Killing horizon is a codimension one null hyper-surface where norm of a Killing vector vanishes. There are six Killing vectors formed by linear combination of functions K ± 0 and K ± ±1 which are bi-linears made out of solutions to Schrödinger equations ψ ′′ − f + ψ = 0 and φ ′′ − f − φ = 0. Out of these six, two of them generated by K ± 0 are periodic and globally defined. If we denote these two normalized solutions as ψ α and φ α , one can show that the norm of the most general combination of the two globally defined Killing vectors may always be brought to the form with h i given in (2.2) and and K is an arbitrary constant, which as we will show below, is equal to the surface gravity by an appropriate choice of normalization for ζ H .
Killing horizons. Geometries in (1.2) have a Killing horizon at the locus ζ H vanishes. This happens at H(r, One can then readily show that Therefore, H = 0 is a null surface. The Killing horizon is located on the two dimensional surface Note that (3.1) has another solution,r 2 H , . (3.7) We will be assuming that r H is the outer horizon, i.e. r 2 H ≥r 2 H for any given x ± .

JHEP07(2014)104
Surface gravity. Given Killing vector field (3.2) one can compute the surface gravity κ: Performing the straightforward analysis shows that κ = K, which, as expected, is a constant. We note that, as the BTZ example below indicates, the value of surface gravity κ is specified by thef ± (through normalized wavefunctions ψ and φ).
Since the surface gravity is non-zero, the Killing horizon, if exists, is a bifurcate horizon. The cases with vanishing surface gravity, the extremal horizon, is a different family which will be discussed in section 4.
BTZ example. When f + and f − are positive constants, K ± are also constant. With our normalization K ± = κ/(2 f ± ), and horizon is at H BT Z = 0 where, The standard BTZ normalization for the ζ H is such that, In the extremal BTZ case, f − = 0 (or f + = 0) and it is readily seen that surface gravity vanishes, κ = 0, and Ω = 1 (or Ω = −1).

Is the bifurcate Killing horizon compact and simply-connected?
As mentioned the Killing horizon, which we showed it always exists, is generically a bifurcate horizon. We then have a "bifurcation curve" which is a one dimensional curve on the r 2 = r 2 H surface. The vectors ζ H and ∇H should be normal to this curve. Therefore, we can already find the vector along (tangent to) the bifurcation curve. One can show that there is another linear combination of globally defined Killing vectors generated by K + 0 and K − 0 , denoted by T , which is tangent to the bifurcation curve. This Killing vector field, as in the BTZ case, remains spacelike everywhere outside horizon, as long as |ζ H | 2 ≤ 0. A straightforward computation shows that |T | 2 = ℓ 2 at the horizon. 5 The Killing vector T is tangent to the bifurcation curve and the necessary (but not sufficient) condition for having an event horizon is T to be periodic. Periodicity of T is guaranteed iff K ± 0 are periodic. This latter follows from the Floquet theorem.
To explore presence of an event horizon, we first analyze compactness and simplyconnectedness of the possible Killing horizon. In what follows we have enumerated the conditions for having a compact or simply-connected bifurcation curve: 1. H = 0 (3.4) has acceptable solutions. That is, we need to require everywhere on the x ± plane. 5 The fact that ζH remains time-like everywhere outside horizon is particular to BTZ (and possibly other asymptotic AdS) black holes. For the generic stationary asymptotic black holes, like e.g. Kerr-Neumann black hole, ζH is asymptotically space-like.
3. Generators of K ± 0 and also r 2 H should be periodic functions on x ± plane, i.e. 4. Bifurcation curve should be compact or simply-connected. Suppose that the bifurcate curve is given by where s parameterizes the one dimensional bifurcation curve. We choose the range of s such that s ∈ [0, s 0 ]. As discussed above, the tangent to this curve T is along a linear combination of K ± 0 Killings (which is a space-like Killing vector at r = r H given in (3.6)). Explicitly, where the derivatives are computed on the horizon and ξ µ are components of T and given in (2.2) with Length of horizon is then given as Therefore, the compactness condition of the bifurcate curve translates into the finiteness of s 0 . This condition and the connectedness condition of bifurcation curve is satisfied if ξ µ are finite everywhere at the horizon. This means that the denominator in (2.2) should not vanish at the horizon, or There are, however, two important exceptions to the above condition: (a) r 4 H = ℓ 4 f + f − but K ′′ ± = 0 so that the numerator in (2.2) also vanishes at r H . This latter can only happen for the constant f ± , the BTZ case.

JHEP07(2014)104
(b) When f − or f + is vanishing. This latter is the case discussed in [21] and we will come to it in the next section.
So, hereafter we exclude the above two cases and explore nonconstant, nonvanishing f ± cases.
Positivity of r 2 H , noting that ψ and φ are completely independent functions, imply that if both f ± are nonvanishing, (3.16) may be written as where we used the fact that ψ and φ are two independent functions of different arguments. The above may also be written as Nonetheless, (3.19) and (3.17) cannot be simultaneously satisfied because a periodic continuous function should necessarily have points where its derivative changes sign. We hence conclude that the only cases with compact and simply-connected Killing horizon are exactly the two cases excluded above, that is, the constant f ± (the BTZ) case or the degenerate horizon case with f − or f + equal to zero. This is accord with the discussions of [21].

Is the Killing horizon also an event horizon?
So far we showed that geometries in (1.2) generically have a bifurcate Killing horizon and argued that the bifurcate curve is generically noncompact or not simply-connected. In order to qualify as a black hole, however, it is necessary to have event horizon. This is more crucial for the AdS 3 black holes, as all the solutions to AdS 3 Einstein gravity are locally AdS 3 . Unlike the Killing horizon which is a local feature of the geometry, the notion of event horizon is a global property of the geometry, e.g. see [26] for more detailed discussions. For asymptotic AdS spacetimes, like in our case, future (or past) event horizon H + (or H − ) is defined as the boundary of the closure of causal past (or future) of the points on the boundary of the AdS space. The Penrose theorem (see e.g. [26]) stating that the generators of H + (or H − ) have no future (or past) endpoints, also extends to the asymptotic AdS case. In terms of the causal structure and the Penrose diagram this means that the past JHEP07(2014)104 or future event horizon must only intersect the causal boundary in two points (once in the "infinite past" and once in the "infinite future"). If the horizon does not intersect the boundary, this means that one can circumvent the horizon and pass to the other side at finite AdS coordinate time and hence we do not have a black hole.
Intuitively, having an infinite length noncompact (or not simply-connected) bifurcation curve on the Killing horizon may seem to be suggesting that it either does not intersect the AdS boundary or it intersects it many times (in one period of x ± ) and hence, is not an event horizon. In this part we bring arguments that the Killing horizon with noncompact (or not simply-connected) bifurcation curve is not an event horizon. To this end, let us focus on the null geodesics of metric (1.2). Since we are interested in the past (or future) null geodesics ending at the causal boundary of asymptotic AdS, it is more convenient to adopt the coordinate system which makes manifest the time and space part of the 2d boundary surface. These coordinates t, ϕ were introduced in (1.3) in which metric (1.2) takes the following form In this coordinate system equation of the radial (zero angular momentum) null geodesic iṡ where dot denotes derivative with respect to t. The plus (minus) sign corresponds to future (past) oriented geodesic. For our analysis we should focus on the past oriented null geodesics starting from an arbitrary point at the AdS boundary and examine if it reaches and intersects the horizon or not. For the "radial null geodesics" described in (3.21), sinceṙ changes sign at r 4 0 ≡ ℓ 4 f + f − , for having an event horizon we need to require that r 4 H ≥ r 4 0 , or equivalently the necessary condition for having an event horizon is essentially the same condition for compactness and connectedness of the horizon (3.16). (The special case of r 4 H = ℓ 4 f + f − corresponds to either of the two "exceptional cases" discussed in previous subsection, for which we do have event horizon.) In view of the argument above, few comments are in order: • Our discussions above can be readily extended to the more general non-radial null geodesics, noting that addition of angular momentum will add a positive-definite contribution to the left-hand-side of the first equation in (3.21).
• The condition for having event horizon and compact and connectedness of the bifurcation curve on the Killing horizon has come about because f ± generically have x ± dependence and this makes the Killing horizon radius r H (3.6) be different than

JHEP07(2014)104
• r 0 is basically the end of the range for the coordinate system used in (1.2); in particular we note that The above argument completes our prerequisites enumerated in the introduction for viewing generic geometries (1.2) as semiclassical fuzzball microstates of BTZ black holes. This will be discussed further in section 5.

The degenerate horizon case
In the previous section we showed that when f ± functions in metric (1.2) are non-zero and are not constants, the geometry has a bifurcate Killing horizon but not an event horizon. Our argument for nonexistence of event horizon, however, had two exceptions, constant f ± and when one of f − or f + is vanishing. The constant f ± case as discussed corresponds to BTZ black hole which admits event horizon [9,10]. In this section we discuss the other case, by setting f + = 0, and consider metric of the form We start analysis of metric (4.1) by studying its Killing vector fields. The Killing vectors are given by (2.2) but now, K + and K − satisfy The most general solutions for the K + equation is a linear combination of the three solutions K +,−1 = 1 , K +,0 = x + , K +,+1 = (x + ) 2 . The Killing vectors associated with these three solutions form an sl(2, R) algebra. It is explicitly seen that only the generator corresponding to K +,−1 (this is generator of translations along x + direction) is periodic in x + . 7 It is readily seen that the norm of this global Killing vector field, ∂ x + , is zero and hence the geometry has a globally defined null Killing vector field and a Killing horizon at r = 0. This geometry should hence fall into the class constructed and discussed in [21]. One can indeed construct a coordinate transformation which (locally) maps the geometry (4.1) to the one in [21] (through this map, the function f − (x − ) will be mapped onto the function β(x), see eq. (31) of [21]). More detailed analysis of the Killing algebra of the geometries in [21] and their connection to the coordinate system we have adopted in this paper is gathered in the appendix C. 6 Here we have shown that there is no Killing horizon which is also an event horizon and there is still the possibility of having a compact and simply-connected event horizon which is not Killing. To rule out the latter and complete proof of nonexistence of event horizons, the above analysis should be supplemented by a full analysis of the causal structure of the geometries (1.2) which may extend coordinates beyond r0. This latter will be postponed to future publications. 7 One may represent the three K's in (4.3) in terms of solutions to Schrödinger-type equation ψ ′′ = 0, as in (2.5). The solutions of this equation are ψ1 = 1, ψ2 = x + , and hence K+,−1 = ψ 2 1 , K+,0 = ψ1ψ2, K+,+1 = ψ 2 2 . Note that, unlike the generic case with non-zero f+, in this case the K+,−1, and not K+,0, is periodic; for this special case Floquet theorem does not apply.

JHEP07(2014)104
Depending on the compactness of x + , we have two distinct classes of geometries: 1. Compact x + case, where (4.1) corresponds to geometries which are either "semiclassical microstates" (or excitations) of an extremal BTZ black hole whenf − ≥ 0 or to an "extremal" conic space (an AdS 3 particle with equal mass and angular momentum) whenf − < 0. The special case of f − = 0 corresponds to massless BTZ.
The discussions in this case is essentially the same as the generic case of previous two sections, with the important difference that we have degenerate (extremal) Killing horizon. The metric at Killing horizon which is located at r = 0 is 2. Noncompact x + case, while x − is still compact. In this case and for generic nonconstant f − the geometry has a global R × U(1) isometry. In the constant, positive f − case, this geometry corresponds to the Self-Dual AdS 3 Orbifold (SDO) and has a global Sl(2, R) × U(1) isometry [11][12][13]. This geometry does not have an event horizon while it has a Killing horizon at r = 0. The two compact and noncompact x + cases are related through the near-horizon limit: while keeping x − , u and ρ fixed. As was discussed in [27], the above scaling decompactifies x + . On the dual 2d CFT this near horizon limit corresponds to performing a Discrete Light-Cone Quantization (DLCQ) [27].
Since the two cases are related by the near-horizon limit (4.5) which keeps the x − dependent part of the metric (4.1) intact, we may discuss both cases simultaneously. These geometries have three other Killing vectors (local isometries) associated with three solutions of (4.2) for K − . These three Killing vectors are specified through the solutions to the Schrödinger-type equation which as discussed in subsections 2.1 and 2.2, form a local sl(2, R) algebra. The Floquet theorem again implies that the generator of the u(1) ∈ sl(2, R), the K 0 , is periodic in x − . This result is in agreement with the analysis in [21]. For more details see appendix C. This the Fourier modes of function f − , which are nothing but the conserved charges associated with a chiral-half of a Virasoro algebra generated by the Brown-Henneaux diffeomorphisms [21,22,27]. 8

Concluding remarks and outlook
In this work we revisited the problem of AdS 3 gravity and took the first step toward its quantization. Our starting point was recalling that all solutions to AdS 3 Einstein gravity are locally AdS 3 and that these geometries can be classified according to their global features (like topology of the asymptotic geometry) and more importantly, by the behavior of the metric in the asymptotic AdS 3 region. The most general geometries with the Brown-Henneaux boundary behavior are geometries in (1.2). These solutions are specified by two "holomorphic" functions f + (x + ) and f − (x − ) which in the standard AdS 3 /CFT 2 dictionary [15,20,23] are related to the energy momentum of the dual 2d CFT.
Here, we used the fact that all the geometries in (1.2) are diffeomorphic to AdS 3 (up to "large" coordinate transformations). Therefore, all the information in these geometries are equally encoded in the large diffeomorphisms which map them to AdS 3 . This viewpoint is complementary to that of standard the Brown and Henneaux [16], where diffeomorphisms with specific boundary conditions, which take us away from the AdS 3 , are considered and analyzed. The two viewpoints are of course closely related as the comparison between set of our Killing vectors (2.2) and the Brown-Henneaux diffeomorphisms indicates. In other words, Brown-Henneaux boundary gravitons are in one-to-one relation with the set of Killing vectors associated to a geometry with given f ± . The diffeomorphisms which map our geometries to AdS 3 are completely specified by the form of the Killing vectors of the geometries. Being locally AdS 3 , there are six such Killing vector fields which form local sl(2, R) × sl(2, R) isometry algebra. So, we focused on studying these Killing vectors.
Further analysis of the geometries in (1.2) and their Killing vectors fields revealed a very interesting feature: generic geometries in this class have always (bifurcate) Killing horizon, which is of course generated by a linear combination of these Killings and is a local feature of spacetime, but do not generically have a compact (finite length) and simplyconnected bifurcation curve. This latter is compatible with the analysis of [21]. Moreover, we argued that these Killing horizons are not event horizons. Our analysis for the latter was presented in section 3.2, but a more thorough analysis is needed to prove that these geometries do not have compact event horizons which are not Killing horizons. This latter is an unlikely possibility if we restrict to geometries which satisfy Brown-Henneaux boundary conditions, as we did, noting that these geometries are specified by the conserved charges uniquely determined through functions f ± . Further analysis on this issue is postponed JHEP07(2014)104 to future works. 9 Stationary black holes, as BTZ black holes, on the other hand, are geometries with event horizon. So, as we discussed in detail our proposal is to view the geometries in (1.2) which are not generically black holes, as "semiclassical" microstates around a BTZ black hole specified by the average of functions f ± (cf. (1.7)). This is a direct realization of the fuzzball proposal [25]: each of these semiclassical microstates, the fuzzball geometries, are specified by two functions f ± or equivalently by their Fourier modes and are superposed/averaged to a BTZ black hole.
This picture is very simple and interesting, but the number of these microstates are not large enough to account for the entropy of the associated BTZ black hole. We need to perform quantization over these semiclassical microstates to get a full quantum description of BTZ black hole microstates and probably then an AdS 3 quantum gravity. 10 Quantization of these geometries is what will be studied in our next publication. However, here, we make some remarks on our vision for this quantization. One may follow either of the two ideas outlined below: 1. Promoting f ± to operators f ± : such that their ensemble average equals their space averagef This quantization proposal is closely related to the one in standard Brown-Henneaux analysis [14-16, 19, 22], where one is prescribed to directly relate Fourier modes of f ± to the Virasoro algebra generators, or equivalently, 3) and c is the Brown-Henneaux central charge (1.6). This quantization proposal is basically the standard Weyl correspondence rule of quantum mechanics applied to AdS 3 gravity within our proposal. With the above choice and for any state in the Hilbert space which form a representation of the above Virasoro, condition (5.1) is automatically satisfied. One should, however, note that these "semiclassical" microstate geometries correspond to "mixed states" (rather than pure states) in the Virasoro representations or in the corresponding dual 2d CFT.
2. Boundary gravitons as "edge-states." Another idea for quantization, which is of course closely related to the previous idea, is to view Brown-Henneaux boundary gravitons as edge-states in a quantum Hall system. This idea is based on the observation that AdS 3 9 We thank James Lucietti and Steve Carlip for discussions on this issue. 10 See the papers by S. Carlip, e.g. [28,29], for related analysis and ideas.

JHEP07(2014)104
Einstein gravity, at least classically, is equivalent to an sl(2, R) × sl(2, R) Chern-Simons gauge theory [1-4, 7, 8]: and A ± µ are sl(2, R) valued gauge fields, which are components of the one-forms with ω a being the sl(2, R) spin connections and e a the dreibein, and κ = ℓ 4G = c 6 . On the other hand, quantum Hall system may also be described by a Chern-Simons theory [30,31]. Chern-Simons theory, and hence the corresponding quantum Hall system is a topological theory and hence the "dynamics" of system is completely determined by the "edge states" describing the boundaries of the quantum Hall droplets, e.g. see [32,33] and reference therein. The idea we would like to put forward is somehow paralleling the same idea as edge-states: the quantum dynamics of AdS 3 gravity is described by its "edge states". In a sense this idea is not new, as notion of Brown-Henneaux boundary gravitons is essentially very similar. 11,12 However, what makes this idea more tractable is the detailed analysis of the Killing vectors of most general semiclassical geometries associated with the Brown-Henneaux boundary gravitons, i.e. geometries in (1.2). As already mentioned, there is a one-to-one correspondence between the Killing vectors of these geometries and boundary gravitons. In addition, as we discussed in this paper, these Killing vectors may be constructed through solutions to the Schrödinger-type equation Ψ (see appendix A for more discussions). The sl(2, R) doublet Ψ is indeed what we would like to identify as the "edge-state" (this makes a very close parallel to the notion of edge state in quantum Hall literature, e.g. see [32,33]). Explicitly, the action which describes the AdS 3 gravity and its edge-states is where S CS is given in (5.4) and where D ± = ∂ ± + A ± (cf. appendix A.1). The idea is then to quantize (5.7). Here, the edge states Ψ and Φ play a fundamental role and boundary gravitons may be recovered through them. Performing the details of this quantization is postponed to our upcoming work. However, before closing we would like to discuss some other interesting and related ideas: 11 It is also inspiring to recall that gravity on AdS space, due to the causal boundary of AdS, is basically behaving like gravity in a box. This box may be thought as the quantum Hall droplet in our analogy. Then the AdS boundary, is the edge of the droplet. 12 See e.g. [34] for some related discussions.
As we discussed in sections 3 and 4 there is an interesting subclass of geometries of (1.2) which correspond to cases with degenerate horizon. These geometries which are obtained by setting f + = 0 were also constructed and discussed in [21]. As discussed in the end of section 4, these geometries unlike the generic f + = 0 case, have generically event horizon, too. As pointed out in section 4, one should modify our semiclassical microstate proposal a little bit to fit these cases. Moreover, this case can come with compact or noncompact x + , where the two are related through the DLCQ procedure in the dual CFT [27]. Since in this case we are dealing with a single function f − and hence a chiral sector of a Virasoro algebra, it would be a very good test ground for examining the quantization proposals outlined above.
Connection to Kerr/CFT. The above picture for quantizing AdS 3 gravity will have direct implication and application in the higher dimensional gravity theories. In particular, it is known that geometry appearing in Near Horizon limit of Extremal black hole Geometry (NHEG) generically develop an AdS 2 throat with Sl(2, R) × U(1) N isometry, for a review see [35]. The Kerr/CFT proposal [36,37], 13 then states that microstates corresponding to the NHEG (and hence to the extremal black hole) is described by a chiral 2d CFT and that states of this 2d CFT are labeled by the asymptotic symmetry group associated with the diffeomorphisms of prescribed boundary conditions on the NHEG background. It would be very interesting to extend the picture for AdS 3 boundary gravitons we presented here to the NHEG boundary gravitons discussed in Kerr/CFT proposal [36,37].

JHEP07(2014)104
The index S on the bracket is chosen to denote the fact that this bracket is induced from the normalization condition of the Schrödinger equation (2.6).
Let us now compute the algebra of Killings K a , a = −, +, 0 generated by ψ 1 , ψ 2 , as given in (2.5). A simple calculation shows that The above is clearly the sl(2, R) algebra. In fact one can show that the above bracket between the Killings K a is nothing but the Lie bracket of these vectors, see [14] for further discussions.

A.1 Relation to 2-d Dirac equation
From second order differential equations (2.4) on ψ and φ, one can get first rank equation similar to Dirac's equation. Let us introduce doublets Ψ and Φ where ψ α and φ α are solutions to (2.4). One can then show that Ψ and Φ satisfy following equations where τ a are 2 × 2 generators of sl(2, R) given in (A.9), n a is such that (n a τ a ) 2 = 1 and (with similar relations for both + and − cases). As we see the gauge field is specified by a single function B (the sl(2, R) gauge field) which in turn is fixed by the function in the metric f . The sl(2, R) part of the gauge field is along a unit vector n a , which specifies a point on a unit radius AdS 2 space [38].
With the above one can write the explicit solution for (A.5) as where Ψ 0 and Φ 0 are two arbitrary constant Sl(2, R) doublets and P denotes the path ordering.
A.2 Representation of sl(2, R) in terms of Ψ-bilinears One can give a representation of the algebra of Killings (A.3) using the Ψ-doublets. To this end we choose the (0, −, +) basis for sl(2, R) and note that in 2 × 2 representation of sl(2, R) algebra, generators take the form

JHEP07(2014)104
The sl(2, R) metric g ab is then {τ a , τ b } = 2g ab 1. One can represent the Killing generators K a given in (2.5) in a more inspiring and formal way as is an sl(2, R) doublet andΨ  Although Hill's equation for a given periodic Q(x) has periodic solutions for specific (discrete set) of λ's, periodicity of Q(x) does not imply periodicity of ψ 1 , ψ 2 . Nonetheless one can show that which implies "quasi-periodicity" condition on ψ's, stated in Floquet theorem: Solutions of Hill's equation satisfy the following Floquet behavior, and e 2iπν , −1/2 < Re(ν) ≤ 1/2 is the eigenvalue of Wronskian matrix ν is called the Floquet exponent and one can show that 2 cos(2πν) = ψ 1 (2π) + ψ ′ 2 (2π).
A useful indicator of solutions of Hill's equation is the discriminant ∆(λ), It has been shown that ∆ is an entire function of order 1/2 of λ. For a given Floquet exponent ν, ∆(λ) − 2 cos(2πν) = 0 has infinitely many roots λ. Conversely, for a given λ, one may compute ν once we have ∆(λ). The discriminant ∆ can be expressed as an infinite determinant involving the Fourier coefficients of Q(x) [39].

B.1 Periodic solutions, spectrum of eigenvalues and (in)stability intervals
Although solutions of Hill's equation for generic "eigenvalues" λ are not periodic, there are specific values of λ for a given Q(x) for which we have periodic solutions. It is readily seen that periodic solutions are those with ν = 0 or ν = 1/2. It has been shown that 1. Eigenfunctions which are periodic with period 2π correspond to ν = 0 case and form a discrete set. Let us denote them by ψ k , k = 0, 1, · · · , i.e. ψ k (x) = ψ k (x + 2π) and the corresponding eigenvalues by λ k . We choose k = 0 for the "ground state" with smallest value of λ.

2.
The explicit values of eigenvalues λ k is completely specified, and can in principle be computed, given the potential Q(x), through the "Hill's determinant." 3. For generic "potential" Q(x), we have an infinite number of periodic solutions. However, there are specific Q(x) where there are finite number of periodic solutions [40].
4. The ground state ψ 0 has no zeros and χ = ψ ′ 0 ψ 0 is a smooth function averaging to zero (in the [0, 2π] range). One can then show that [41] That is, the ground state energy λ 0 is always negative and has a lower bound.
6. It has been proved that for every 2π periodic solution there is a 4π periodic solution, except for the "ground state" and that the eigenvalues are ordered in the following way:
Connection to our Schrödinger eq. From the above form one learns that a coordinate transformation like x − = γdx + K(β, r) , x + = v ℓ + H(β, r) , (C.5) relates (C.1) to (4.1), where the large r expansion of K and H functions is given in eq. (30) of [21]. The important and interesting point is then how the function γ(x) and our Schrödinger equation wavefunctions are related.
Here is the answer: In other words, γ is directly related to the log-derivative of our wavefunction. It is intriguing that the same expression appears in our analysis for norm of the Killing vectors (and location of horizons).
The Killing vectors. Metrics (C.1) and (4.1) are indeed the same geometry written in two different coordinate systems. Since the coordinate transformations which map the two are complicated and hence not illuminating, we find it more useful to compare the Killing vectors fields in these two coordinate systems. As all locally AdS 3 geometries, metrics (C.1) or equivalently (4.1) have local sl(2, R) × sl(2, R) isometries. To see this explicitly we write the six Killing vector fields in both coordinate systems.

JHEP07(2014)104
where K a are as given in (2.5), explicitly and φ α are the two normalized linearly independent solutions to φ ′′ − f − φ = 0 with φ ′ 1 φ 2 − φ 1 φ ′ 2 = 1. As one can explicitly see from (C. 13), K x 0 is periodic in x while K x ± are not periodic. This is in perfect agreement with our earlier discussions on the solutions for the Hill's equation and the Floquet theorem, which states that K 0 − in (C.15) is periodic, while K ± − are exponentially decaying or growing with x.
To summarize, the metric (C.1) (with noncompact or compact v) with γ = 0 has global U(1) v × U(1) x isometry as the metric in (4.1) (with noncompact or compact x + ). In the noncompact x + (or v) case these geometries correspond to Brown-Henneaux boundary gravitons of AdS 3 self-dual orbifold, while when x + (or equivalently v) is compact, the geometry corresponds to extremal BTZ geometry excited with Brown-Henneaux boundary gravitons.
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.