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The Structure of AdS Black Holes and Chern Simons Theory in 2+1 Dimensions

We study anti-de Sitter black holes in 2+1 dimensions in terms of Chern Simons gauge theory of anti-de Sitter group coupled to a source. Taking the source to be an anti-de Sitter state specified by its Casimir invariants, we show how all the relevant features of the black hole are accounted for. The requirement that the source be a unitary representation leads to a discrete tower of states which provide a microscopic model for the black hole.

a manifold M with topology R × Σ, where Σ is a two dimensional space. We take the theory to be an explicit realization of the Mach Principle, so that in the absence of sources the field strengths vanish and the topology is trivial (no punctures). One can then associate non-trivial topologies to the presence of sources [5,7,8]. In this scenario, the physical (metrical) space-time is the output of such a gauge theory and should not be confused with the manifold M. The physical space-time is related to a manifold M q the points q A of which are one of the canonical variables (0 + 1 dimensional fields) of the source(s) [7,8]. The presence of sources in M affect not only the topology of M but also the structure of M q as the emerging space-time. It is therefore no contradiction to state that a Chern Simons theory in M (with a source) leads to the black hole solution in M q (with no source).
One of the notable advantages of the Chern Simons approach is that it allows us to express the asymptotic observables of the theory in terms of the properties of the sources. To implement this idea, we must identify a localized source (particle) with an irreducible representation of the gauge symmetry group [8]. For the present problem, this will amount to relating the asymptotic observables of the BTZ black hole to the Casimir invariants of an AdS state coupled to the Chern Simons action. We will show that the emerging space-time will naturally arise from such a theory and will have all the ingredients necessary for the AdS black hole [9]. These include, in particular, the discrete subgroup underlying the identifications. Moreover, the horizon radii of the BTZ solution are complicated functions of the familiar AdS labels M and J, which are commonly referred to as "mass" and "angular momentum", respectively. One might wonder if there is a group theoretic or some other explanation for their functional form. We will show that they are alternative labels for an AdS state and arise naturally from the maximal compact subgroup of the AdS group via induced representations.
An important consequence of the Chern Simons formulation, which we will address in this work is the extent to which the potential quantum aspects of the formalism will influence the choice of the AdS representations. As mentioned above, we take the sources which couple to the Chern Simons action to be AdS states, so that, to have a unitary quantum theory, these states must be unitary representations of the AdS group. One of the remarkable byproducts of this requirement is that the ground state and the excited states of the black hole form a discrete spectrum. Therefore, the Chern Simons theory described below provides a microscopic model of the black hole structure, which appears to be distinct from previous suggestions [10,11].
In Section 2, we review the properties of AdS space and algebra in a form which will be used in subsequent sections. In Section 3, we express the Chern Simons action for the AdS group in an SL(2, R) × SL(2, R) basis. Section 4 is devoted to the interaction with sources. Among other things, we discuss the important role played by the constraints in relating the invariants which label the sources to the asymptotic ovservables of the coupled theory. In Section 5, we explore the consequences of requiring that a source be represented by a unitary representation of the AdS group.
We will show that one of the hitherto unexplained features of the BTZ black hole emerges from this requirement. In Section 6, we show how the black hole space-time emerges from the Chern Simons gauge theory described in sections 3 through 5. In particular, we show how such features as the periodicity of the angular coordinate and the discrete identification group are accounted for. Section 7 is devoted to further discussion of the results and their possible relevance to black holes in other spacetimes dimensions.

Anti-de Sitter space and algebra
The anti-de Sitter space in 2+1 dimensions can be viewed as a subspace of a flat 4-dimensional space with the line element It is determined by the constraint where l is a real constant . The set of transformations which leave the line element invariant form the anti-de Sitter group SO(2, 2). It is locally isomorphic to SL(2, R)× SL(2, R) or SU(1, 1) × SU(1, 1). From here on by anti-de Sitter group we shall mean its universal covering group. The AdS algebra consists of the elements M AB satisfying the commutation rela- With A = (a, 3) and a = 0, 1, 2, we can write the algebra in two more convenient forms: where Then, the commutation relations in these bases take the form, respectively, The Casimir operators look simplest in the latter basis: In the other bases, they have the form, We will use the same symbols for operators and their eigenvalues. An irreducible representation of AdS group can be labeled by the eigenvalues of either the pair (M, J) or the pair (j + , j − ). For our applications, it is often advantageous to use a third set of labels which we denote by (H,S). They correspond to the maximal compact subgroup SO(2) × SO(2) of SO(2, 2), which is generated by J 0 and Π 0 . The labels (H, S) are a natural choice from the point of view of the theory of induced representations. This can be seen from the comparison with the more familiar situation in the Poincaré group which can be obtained from anti-de Sitter group in the limit l → ∞. From here on, we will use the labels, (j + , j − ), (M, J), and (H, S) interchangeably. The last two are related to each other according to Note that in order for M to assume negative values, H and S must, in general, be complex.
To see the relevance of H and S to the BTZ solution, let us express H and S in terms of the labels M and J by inverting Eqs. (10). We obtain For M > 0 and |J| ≤ lM , H and S are thus proportional to the horizon radii, r ± , of the BTZ black hole [1]:

Connection and the Chern Simons action
We begin by writing the connection in SL(2, R) × SL(2, R) basis where Eqs. (14) and (15) should be viewed as definitions of e and ω in terms of SL(2, R) connections. The covariant derivative will have the form Then the components of the field strength are given by For a simple or a semi-simple group, the Chern Simons action has the form where Tr stands for trace and We require the 2+1 dimensional manifold M to have the topology R × Σ, with Σ a two-manifold. So, The Chern Simons action with SL(2, R) × SL(2, R) gauge group will take the form Here the quantities a ± are, in general, arbitrary coefficients, reflecting the semisimplicity of the gauge group. Up to an overall normalization, only their ratio is significant. It was pointed out by Witten [5] that in the free Chern Simons theory the choice a − = −a + would make the action proportional to Einstein's action in M by imposing a metric structure on it. Similarly, the choice a − = a + would give an "exotic" term. He also pointed out that, in our notation, for generic values of these coefficients, the classical equations of the free theory in M remain unchanged. In a quantum theory [5], the two terms in the action will have a relative arbitrary coefficient.
It would be tempting to choose the first possibility on grounds of familiarity, among other things. However, that would be an unnatural choice from the point of view pursued here. This is because the space-time which emerges from this theory is not the manifold M but a manifold M q corresponding to one of the canonical variables (0 + 1 dimensional fields) of the source which will be coupled to the Chern Simons theory in the next section. So, the space-time is a secondary concept which emerges from the gauge theory, and Einstein's action in M plays no direct role in it. Moreover, in the presence of a source (or of sources), anyá priori choice of the coefficients a ± reduces the class of allowed holonomies, so that even the classical theory coupled to sources will be affected by such a choice. For these reasons, we will keep the coefficients a ± as free parameters in the sequel, so that we can generate the correct holonomies for solutions both ouside and inside the horizon.

Under infinitesimal gauge transformations
the gauge fields transform as More specifically, As we have stated, the manifold M has the topology R × Σ with R representing x 0 . Then subject to the constraints the Chern Simons action for SO(2, 2) will take the form

Interaction with sources
Following the approach which has been successful in coupling sources to Poincaré Chern Simons theory [8], we take a source for the present problem to be an irreducible representation of anti-de-Sitter group characterized by Casimir invariants M and J (or H and S ). Within the representation, the states are further specified by the phase space variables of the source Π A and q A , A = 0, 1, 2, 3, subject to anti-de Sitter constraints. For illustrative purposes, let us consider first the interaction term for a special case which is the analog of the Poincaré case [8] with the intrinsic spin set to zero.
where C is a path in M, τ is a parameter along C, and the covariant derivative D τ is given by The first term in this action is the same as that given in reference [4]. The second term ensures that q A (τ ) satisfy the AdS constraint. It is not the manifold M over which the gauge theory is defined but the space of q ′ s which give rise to the classical space-time. The last two constraints identify the source being coupled to the Chern Simons theory as an anti-de Sitter state with invariants j + and j − . These constraints are crucial in relating the invariants of the source to the asymptotic observable of the coupled theory. In this respect, our action differs from that given in reference [4]. Although the word "constraints" was mentioned there in connection with this action, they were not explicitly stated or made use of in the sequel.
Using the standard (orbital) representation of the generators we have Here L AB are c-number quantities transforming like M AB . Breaking up this expression into SL(2, R) × SL(2, R) form just as was done M AB , we get So, the action I 1 can be written as In this expression L ± a play the role of (c-number) generalized orbital angular momenta. iF, in addition, the representation carries generalized intrinsic (spin) angular momenta, then L ± a would have to be replaced by J ± a , respectively, where It is now clear how the interaction term I 1 can be generalized to the case when S ± a = 0. We simply replace L ± a with J ± a in I 1 to get This expression is identical in form to that given by Eq. 26. But now the generators are not limited to the form given by Eq. 28. It can be expressed in a form in which the SL(2, R) × SL(2, R) structure of the gauge group is transparent: In this expression J ± a play the role of c-number generalized angular momenta which transform in the same way as the corresponding generators and which label the source. iF there are several sources, an interaction of the form given by Eq. 35 must be written down for each source.
It is well known that for a Poincaré state with mass m 2 > 0, there is a (rest) frame in which, e.g., the momentum vector takes the form Similarly, in the present case, there is a frame such that when, e.g., the c-number quantity J ±a J ± a > 0, we have In this gauge, SL(2, R) × SL(2, R) symmetry reduces to SO(2) × SO(2). One can use similar methods to choose a gauge in which the residual symmetry is, e.g., SO(1, 1) × SO(1, 1). Combining, the interaction term I s with the Chern Simons action I cs , we get the total action for the theory; In this theory, the gauge fields A ± µ and the phase space variables q A are smooth functions on the manifold M. Gauge transformations on the former, which are components of the connection in the principal SO(2, 2) bundle, induce appropriate gauge transformations on the associate bundle to which the latter belong. It is easy to check that the components of the field strength still vanish everywhere except at the location of the sources. So, the analog of Eqs. 24 becomes In particular, when η ab J ± a J ± b > 0, we get, in the special (rest) frame All other components of the field strength vanish. We thus see that because of the constraints appearing in the action given by Eq. 34, the strength of the sources corresponding to the maximal compact subgroup of the gauge group become identified with their Casimir invariants. These invariants, in turn, determine the asymptotic observables of the theory. Since such observables must be gauge invariant, they are expressible in terms of Wilson loops, and a Wilson loop about our source can only depend on, e.g., j + and j − . From the data on the manifold M given above, it is possible to determine the properties of the emerging space-time. To this end, we note that in the gauge in which Eq. 39 holds, the only non-vanishing components of the gauge potential are given by where θ is an angular variable. As an example, consider the case of a + = a − = 1. Then, using Eqs. 14 and 15, the non-vanishing components can also be written as, Although these are components of a connection which is a pure gauge, they give rise to non-trivial holonomies around the source. More explicitly, we have Here, γ is a loop around the source, which can be represented as a map from the circle to the manifold M, i.e., γ : S 1 → M with γ(σ + 2π) = γ(σ). These holonomies are not gauge invariant [2] and transform by conjugation under SO(2) × SO (2)

Restriction to unitary representations
We have indicated that our sources transform as irreducible representations of the AdS group. From purely classical considerations, the choice between unitary and non-unitary representations might not seem to be relevant. But to allow for the possibility of quantizing the Chern Simons theory consistently, we will require that our sources be represented by unitary representations of AdS group. As we shall see, this requirement will also have interesting consequences for the classical space-time which emerges from this theory.
Since the AdS group in 2 + 1 dimensions can be represented in the SL(2, R) × SL(2, R) form, we can construct the unitary representations of SO(2, 2) from those of SL(2, R). We will assume that the reader is familiar with the representation theory of SL(2, R). Here, we will state a few facts relevant to its unitary representatons [12].
More information about these representations can be found in papers listed in reference [12] and those cited therein. The states in an irreducible representation of SL(2, R) are specified by the eigenvalues of its Casimir operator j 2 (see Eq. 8) and, e.g., the element J 0 , where we have suppressed the super-(sub)scripts ± distinguishing our two SL(2, R) 's. Thus, we have In these expressions, Φ is, in general, a complex number, F is a fraction, and m is an integer. It is well known that SL(2, R) has four series of unitary representations [12], all of which are infinite dimensional. For the present application, we choose the discrete series in which each irreducible representation is an infinite tower of states for which the eigenvalues of J 0 are bounded from below. That is, So, for this series, in the notation of section 2, the eigenvalues of the Casimir invariants of SL(2, R) × SL(2, R) can be written as, It follows that the infinite set of states can, in a somewhat redundent notation, be specified as |j 2 ± , F ± + m ± >; m ± = 0, 1, 2, ...
Clearly, the integers m ± are not necessarily equal. Using these states, we can construct the discrete series of the unitary representations of SO(2, 2). A typical state will have the following labels: As a prelude to identifying the labels M and J with the corresponding labels in the BTZ solution in the next section, let us consider the physical restrictions imposed on F ± . It is clear from Eqs. 11 through 13 that, up to proportionality constants, we want to identify r 2 ± with H 2 and S 2 , respectively. To do so, we must choose positive square roots of H 2 and S 2 since the radii are intrinsically positive quantities. We also see from these equations that to have real non-zero horizon radii, we must have M > 0. Then, since the two SL(2, R) 's appear symmetrically in the formalism, we must take j 2 ± to be real and positive. Then, Eq. 46 requires that F ± > 1. Once this condition is satisfied, it can be seen from Eq. 9 that |J/l| ≤ |M|, as required in the BTZ formalism. The extreme case corresponds to j 2 − = 0. This, in turn, requires that F − = 1. We note that similar statements also hold for the Euclidean version of the AdS space, where the symmetry group becomes SO (1, 3). So far we have discussed the unitary discrete series of SL(2, R), and subsequently of SO(2, 2), for which the eigenvalues of J ± 0 are bounded from below. We have seen that they are suitable candidates for the microstates of the AdS black hole. Of the other three series of unitary representations of SL(2, R), the principal series are characterized by eigenvalues of the Casimir operator which are negative. Using the results of section 2, it is easy to show that they lead to negative values of M and to complex values of the horizon radii. On this basis alone, we consider them not relevant to the description of the black holes. The other two series are the supplementary series and the discrete series for which Φ and F are equal and negative. They cannot be ruled out on the basis of the criteria discussed above. However, for these series, as well as for the principal series, the eigenvalues of the operators J ± 0 are not bounded from below. In analogy with real 3 + 1 dimensional world, if we identify the corresponding microstates as physical degrees of freedom, then we will have to identify J ± 0 with some physical observables. It is difficult to conceive of a physical observable with infinitely large negative eigenvalues. Until such an observable could be justified on physical grounds, we seem to be limited to the first discrete series discussed above.

The black hole space-time
To see how the space-time structure emerges from our anti-de Sitter gauge theory, we follow an approach which led to the emergence of space-time from Poincaré [8] and super Poincaré [13] Chern Simons gauge theories. We have emphasized that the manifold M is not to be identified with space-time. But the information encoded in it and discussed in section 4, is sufficient to fix the properties of the emerging space-time. To this end, let us consider a manifoldM q satisfying the AdS constraint where Λ = cosmological constant. In fact, our SL(2, R)×SL(2, R) formulation allows us to takeM q to be the universal covering space of the AdS space. As we shall see, the emerging space-time is the quotient ofM q by the discrete subgroup Γ discussed in section 4. Moreover, the source coupled to the Chern Simons action is an AdS state characterized by the Casimir invariants (M, J) or, equivalently, (H, S). To parametrizeM q consistent with the above constraint, consider a pair of 2-vectors, q t = (q 0 ,q 3 ) = f 2 + l 2 cos(t/l), f 2 + l 2 sin(t/l) where f = f (r), with r a radial coordinate which for an appropriate f (r) will become the radial coordinate appearing in the line element for the BTZ black hole. As far the constraint given by Eq. 49 is concerned, the functional form of f (r) is irrelevant.
The parameters φ and t/l are both periodic. We will keep φ periodic throughout. However, since we are takingM q to be universal covering space of AdS space, we do not have to , and we will not, identify t with t + 2πl. With or without this identification, the vectors q A parametrized in this fashion do not behave in the same way as the vectors in the manifold M when they are parallel transported along a loop encircling the source. Computing the line element in terms of the parameters (t/l, r, φ), we get where "prime" indicates differentiation with respect to r.
Anticipating the results to be given below, let us compare this line element with that for the BTZ black hole [1].
If we identify the labels M and J with the Casimir invariants of an irreducible representation of the AdS group as discussed in the previous sections, we see that the line element given by Eq. 52 corresponds to an irreducible representation with J = 0 and M = −1. Such a state will not correspond to any of the series of the unitary representations of the AdS group discussed in the previous section. Moreover, as we have noted in connection with Eqs. 11 and 12, for these values of J and M, the invariant H is pure imaginary. This, in turn, implies that the quantities r ± will also be imaginary. Thus, we can interpret the line element in Eq. 52 as a special form of the BTZ line element which has been "Wick rotated" into the imaginary axis in the complex H space. In this form, the consequences of the residual gauge transformations involving H and S, or r ± , which we will perform below onq A (τ ) become very similar to those performed in the Poincaré [7,8] Chern Simons gravity. We must keep in mind, however, that in the end, we must Wick rotate the results back to the real r ± axes so that the source coupled to the Chern Simons theory would belong to a unitary representation and that the resulting horizon radii would be real. We thus see that the choice of a unitary representation has interesting classical consequences. With these issues in mind, we want to obtain the space-time manifold M q by performing appropriate gauge transformations onM q . Although the original theory was invariant under SL(2, R) × SL(2, R) gauge transformations, we have already reduced this symmetry by choosing to work in a gauge in which Eq. 39 holds. In fact, the left over symmetry is just SO(2) × SO(2) generated, respectively, by J 0 and Π 0 , or, equivalently, by J ± 0 . So, identifying the parameters φ and t/l, respectively, with each SO(2), consider the local gauge transformation q ′ φ (φ) = e i r + l φJ 0 q φ (φ) It leaves q t (t/l) invariant. Then, since φ is 2π periodic, The level structure exhibited in this model is reminiscent of the bound state structure familiar from atomic physics except that the the energy of the ground state is positive. In this respect, we note that in the BTZ solution and the subsequent works the labels M and J have been identified as "mass" and "angular momentum". On the other hand, from the point of view of induced representations, it is the labels arising from the maximal compact subgroup, in this case SO(2) × SO (2), which are more suitable for such designations. In other words, it is the eigenvalues of Π 0 and J 0 which we identify as "energy" and "spin", respectively.
The general features of the formalism developed in this work in 2 + 1 dimensions are applicable to black holes in any dimension. A typical black hole is specified in terms of its asymptotic observables. If we identify these observables with the Casimir invariants of the asymptotic symmetry group, usually a noncompact group, then the corresponding Hilbert space could serve as a microscopic model for the black hole. It remains to be seen whether such models, and their modifications to take into account the fact that the symmetry involved here is a local gauge symmetry and not just a a global one, are sufficiently realistic.