0 1

CERN-TH/97-330

KUL-TF-97/33

hep-th/9711138

Microscopic derivation of the Bekenstein–Hawking entropy formula for non-extremal black holes

Konstadinos Sfetsos Theory Division, CERN

CH-1211 Geneva 23, Switzerland

Kostas Skenderis Instituut voor Theoretische Fysica, KU Leuven

Celestijnenlaan 200D, B-3001 Leuven, Belgium

Abstract

We derive the Bekenstein–Hawking entropy formula for four- and five-dimensional non-supersymmetric black holes (which include the Schwarzchild ones) by counting microscopic states. This is achieved by first showing that these black holes are U-dual to the three-dimensional black hole of Banados–Teitelboim–Zanelli and then counting microscopic states of the latter following Carlip’s approach. Black holes higher than five-dimensional are also considered. We discuss the connection of our approach to the D-brane picture.

CERN-TH/97-330

KUL-TF-97/33

November 1997

## 1 Introduction

Black holes are one of the most fascinating objects in general relativity. Their existence has profound implications for gravity in both the classical and the quantum regime. Black hole quantum mechanics provides a window into strong coupling quantum physics by raising a set of puzzles and questions that any consistent quantum theory of gravity should solve. The discovery that the black-hole laws are thermodynamical in nature [1] implies that there should be an underlying statistical description of them in terms of some microscopic states. In addition, black holes can evaporate [2], which leads to the “information loss paradox”. These questions, to a large extent, remained unanswered for more than twenty years.

String theory claims to provide a consistent theory of gravity. One would therefore expect that string theory provide answers to these questions. The strong coupling nature of black-hole physics, however, requires an understanding of non-perturbative string theory that was not available until recently. The situation has changed dramatically during the last few years. The duality symmetries have led to a new unified picture and provided a handle into strong coupling physics [3]. The discovery of D-branes [4, 5] has led to remarkable progress in the understanding of the physics of extremal black holes. In particular, it led to identification and counting of microstates for this subset of black holes [6, 7]. The result was in exact agreement with the Bekenstein–Hawking entropy formula. The idea behind these computations was to construct a D-brane configuration with the same quantum numbers as the corresponding black hole we are interested in. The counting of states is then performed at weak coupling, where the D-brane description is valid. The BPS property of these configurations implies that the number of states remains unchanged as the string coupling grows. One, therefore, can extrapolate these results to the black-hole phase. In this way states were counted for extremal and black holes. Near-extremal black holes were also studied [7, 8]. The absence of supersymmetry, however, makes these results less rigorous. For the same reason (i.e. absence of supersymmetry) the physically most interesting case, namely the case of non-extremal black holes, is untractable in this framework. Let us mention, however, that a natural extension of these ideas, as formulated in the correspondence principle of Polchinski and Horowitz [9] (for earlier ideas see [10]) does yield the correct dependence of the entropy on the mass and the charges, even if it does not provide the numerical coefficient. Recently, similar results for non-extremal black holes were obtained in [11] using the M(atrix) formulation [12] of M-theory.

Another important development in the understanding of the statistical origin
of the black-hole entropy (that actually preceded the D-brane
developments) was Carlip’s derivation [13] of the
Bekenstein–Hawking
entropy formula for the three-dimensional black hole
of Banados–Teitelboim–Zanelli (BTZ)[14].
The latter solves Einstein’s equations in the presence of a negative
cosmological constant and is, therefore, asymptotically anti-de Sitter.
Soon after its discovery it was shown that the BTZ black hole is actually an
exact solution of string theory [15, 16], namely that there is an
exact conformal field theory (CFT) associated to it.
Physics in three dimensions is significantly simpler
than in higher dimensions. In particular, three-dimensional gravity
can be recast as a Chern–Simons theory [17, 18]. If the space has
a boundary then the Chern--Simons theory induces a WZW action in this
boundary. The latter describes would-be degrees of freedom that become
dynamical because certain gauge transformations become inadmissible
due to boundary conditions. Carlip has shown that these degrees of freedom
correctly account for the Bekenstein--Hawking entropy of
the BTZ black hole.^{1}^{1}1The idea that only physical degrees of freedom
defined in a “stretched” horizon may
account for the black hole entropy has been advocated in [19].
In a string theory context it was put forward by A. Sen [20],
in order to reconcile the Bekenstein–Hawking entropy
for extremal electric black holes,
with the entropy of elementary superstring excitations.
It is important to realize that
this result is valid both at extremality and away
from it. However, the method used seems very particular to three
dimensions (see, however, [21]). Notice also that all D-brane
results are for black holes of dimension higher than three.
The main reason for this is that in constructing a solution
out of D-branes one usually restricts oneself to at least three overall
transverse directions, and three-dimensional space-time has two transverse
directions. If the overall transverse directions are
less than three, the harmonic functions appearing in the
D-brane configuration are not bounded at infinity.

To summarize:the D-branes techniques can be used to derive the Bekenstein–Hawking entropy for and supersymmetric black holes, whereas Carlip’s approach is not restricted to supersymmetric black holes, but it seems to apply only to ones. We shall show in this article that one can use the latter to study non-extremal and black holes and, in particular, we will derive the Bekenstein–Hawking entropy formula associated to them with the correct numerical coefficient. Our considerations also apply to higher-dimensional black holes, although we have no derivation of the Bekenstein–Hawking entropy formula in these cases.

Over the last few years a new unifying picture of all five string theories and eleven-dimensional supergravity has emerged [3]. A central rôle in these developments has been played by the various duality symmetries. It is now believed that there exist an underlying master theory, the M-theory, that has all string theories and eleven-dimensional supergravity [22] as special limits. The dualities symmetries can be viewed as some kind of gauge symmetry of this theory. Physical quantities should be “gauge-invariant”, i.e. U-duality-invariant. Choosing one configuration among all its U-duals to describe a physical system corresponds to choosing a particular “gauge”. As in usual gauge theories, some gauges are preferable for answering certain questions than others. We shall show below that the and black holes are U-dual to the BTZ black hole (for related work, see [23]). One may, therefore, choose the “BTZ gauge” in order to answer certain physical questions. In particular, we shall address in detail the question of the statistical origin of the entropy.

The BTZ black hole (for ) is non-singular. One may, therefore, argue that the singularities in the and black holes are “gauge” artefacts. In addition, the fact that the BTZ black hole is asymptotically anti-de Sitter and the and black holes are asymptotically flat implies that the cosmological constant is a “gauge-dependent” notion. Furthermore, the simplicity of the “BTZ gauge” may make tractable the study of the final state of black holes.

One may wonder at this point how is it possible to connect objects of different dimensionality using the U-duality group. Consider, for concreteness, type-II string theory on a torus. Then, the U-dual group is considered to be the (discretized) version of the global symmetry group of the various maximal supergravity theories obtained from eleven-dimensional supergravity by toroidal compactification in dimensions . Therefore, almost by definition, the dualities do not change the number of non-compact dimensions. For static backgrounds, however, one has, in addition to the isometries corresponding to toroidal directions, an extra time-like non-compact isometry. This leads to a larger group. Consider, for instance, the case of compact directions. The T-duality group is . Suppose for a moment that the time is compact with radius . Then the symmetry group would be enlarged to . To see what happens in the decompactification we let become larger and larger while restricting the elements of to the ones that do not mix the coordinates with finite radii with the time coordinate. In the limit the time becomes non-compact and we are left with a subgroup of . The latter is basically a combination of diffeomorphisms of the time coordinate, which involve the compact coordinates and the transformations of the compact coordinates themselves. In particular, this group contains elements that correspond to isometries that are space-like everywhere except at spatial infinity, where they become null. T-dualizing with respect to these isometries changes the asymptotic geometry of space-time [15, 24]. There seems to be a widespread belief that string theory admits only Ricci-flat compactifications. This is, however, not true. We shall exhibit below exact string solutions that correspond to compactifications on and times some torus. T-dualities along the above-mentioned isometries precisely bring us to these compactifications. These, at low energies, reduce to compactifications of supergravity on and times some torus, and therefore connect Poincaré supergravities to anti-de Sitter () supergravities. Compactifying eleven-dimensional supergravity on spheres instead of tori yields the latter. A famous example is the compactification of eleven-dimensional supergravity on , which yields [25] gauged supergravity [26]. In other words, these transformations connect solutions of (or of ) supergravity that correspond to different compactifications. As such, they may connect solutions with different number of non-compact dimensions. From the point of view of M-theory, one may argue that all compactifications of eleven-dimensional supergravity should be on an equal footing. This suggests that the symmetry group of M-theory is actually larger than what is usually assumed. To obtain the full symmetry group one should also consider the various gauged supergravities. The consistent picture that emerges from our discussion of black-hole entropy strongly supports this point of view. We shall, from now on in this article, use the term U-duality transformation to denote the transformation that results from a combination of the usual () T-duality, of the S-duality of type-IIB string theory and of the extra transformations that we mentioned above. We shall also freely uplift results to eleven dimensions.

We shall argue that certain branes, and intersections thereof, are U-dual to supersingleton representations of various anti-de Sitter groups. In this way we have a connection between our considerations and the usual D-brane picture. In particular the branes , and are dual to the supersingleton representation of , , and , respectively. A complete list is given in Table 1 (see section 4). All the configurations listed there (with the addition of a wave, in some cases) are dual to black holes in . Essentially, the duality transformations map the black hole into the near-horizon geometry (with some global identifications). In the present context, however, this is not an approximation.

The picture emerging from our study is that the microscopic
degrees of freedom reside in the intersection region of the various branes,
making up the black-hole configuration.^{2}^{2}2
We need not be
in the weak string coupling limit for our considerations to be
valid. In fact, we will always stay within the black-hole phase,
where the string coupling is strong. Hence, by branes we mean the specific
solutions of the low energy supergravity. In the case where the latter
carry R–R charge, they are the long-distance description of D-branes.
This picture is in harmony with results existing in the literature.
For and extremal black holes,
described by a configuration of D-branes that has a one-dimensional
intersection, the entropy can be obtained by treating the degrees of
freedom as an ideal gas of bosons and fermions in a one-dimensional
compact space. Similar results (but with only qualitative agreement)
hold for near-extremal non-dilatonic black holes [27].
In that case as well, the microscopic description involves a -dimensional
theory, where is the spatial dimension of the intersection
region. Notice, however, that the intersection region
is not a U-duality-invariant notion
since the same black hole can result from different intersections.
For instance, the black holes can be constructed
by either the intersection of an M-theory membrane () with
an M-theory five-brane () and a wave ()
along the common direction, or from the intersection of three
membranes. In the former case the intersection is one-dimensional,
i.e. over a string, whereas in the latter it is zero-dimensional,
i.e. over a point. Let us emphasize that only U-duality-invariant quantities
of the original configuration may be studied in the U-dual
formulation. The entropy of the black hole is such a quantity
and, therefore, can be computed in any dual configuration.

This article is organized as follows. In sections 2 and 3 we concentrate on the and black holes. In particular, in section 2 we show that and non-extremal black holes are U-dual to configurations that contain the BTZ black hole as the only non-compact part. In section 3 we present our microscopic derivation of the Bekenstein–Hawking entropy formula. We follow Carlip’s approach, putting some emphasis on the unitarity issue of the underlying WZW model. In section 4 we discuss the duality between branes and supersingleton representations. In this way we provide a connection between our considerations and the D-brane picture. In section 5 we briefly discuss higher-dimensional black holes as well as intersections of branes (different from the ones discussed in section 2), which yield and black holes. We conclude in section 5. Appendix A contains the eleven-dimensional supergravity configurations that reduce, upon dimensional reduction along a compact direction, to the ten-dimensional solutions used in section 2. Finally, in appendix B we show that higher than five-dimensional black holes are not U-duals to configurations that contain the BTZ black hole.

## 2 U-duality between non-extremal and BTZ black holes

We will show in this section that ten-dimensional configurations, which upon dimensional reduction in an appropriate number of dimensions yield a or a black hole, can be mapped by a chain of dualities and a simple coordinate transformation into a configuration that has as the only non-compact part the BTZ black hole. In particular, the configuration that yields the black hole will be mapped to the configuration , and the one that yields the black hole to . We will show that there is an exact CFT associated to each factor of the final configuration. For this to be true, it is crucial to carry along the gauge fields of the original configuration. After the dualities all the fields acquire their canonical values so that each factor is independently associated to a CFT. In this sense, our considerations also provide exact CFTs associated to and black holes.

The basic mechanism that allows one to map one black hole that is asymptotically flat into other that is asymptotically anti-de Sitter has been discussed in [24]. Here we shall refine this discussion by showing that what was there called shift transformation, is actually a property of the plane-wave solution. Consider the following non-extremal plane-wave solution in dimensions

(1) | |||||

where

(2) |

The coordinate is assumed to be periodic, with radius , so that has the topology of a cylinder. The constant of the off-diagonal part is chosen such that this term vanishes at . One may T-dualize in the -direction to obtain a solution that describes a non-extremal string. In this case, the off-diagonal part of the metric becomes the antisymmetric tensor of the new solution. Our choice of the constant in the off-diagonal part of (1) ensures that the latter is regular at the horizon [28]. We shall call the surface horizon since, as we shall shortly see, the plane wave solution when combined with certain other branes yields and black holes solutions with an outer horizon at . The area of the latter for the solution (1) is equal to

(3) |

where denotes the volume of the unit -sphere.

Let us perform the following coordinate transformation that preserves the cylinder:

(4) |

Requiring that the transformed solution still be of the form (1) and have vanishing off-diagonal part at like (1), and that the asymptotics be different, uniquely fixes to

(5) |

One obtains^{3}^{3}3
In the extremal limit the transformation (4) and the
metric (6) appear to be
singular. In this case, we have to rescale the coordinates and as
and . After taking
the limit in such a way that the charge
is kept fixed, we obtain a well-defined transformation (4)
with , and arbitrary.
The metric (6) has also a well-defined limit.
(with the primes in and dropped)

(6) |

where now

(7) |

Notice that the radius of is now equal to .
We shall call the transformation (4) the shift
transformation.^{4}^{4}4
The definition of the shift transformation is not the same
as the one employed in [24]. There the shift
transformation acted on the fundamental string solution
and it was a combination of the shift transformation as defined in
this article and T-dualities.
One easily checks that the area of the horizon (i.e. of the
surface )
is still equal to (3).
We therefore conclude that the shift transformation does not change the
area of the horizon.

### 2.1 black holes

Consider the solution of type-IIA supergravity that describes
a non-extremal intersection^{5}^{5}5
All configurations studied in this article are built according to the
rules of [29]. In the extremal limit they
are supersymmetric,
and they are constructed according to the intersection rules
based on the ‘no-force’ condition [30].
of a solitonic five-brane ()
a fundamental string () and wave () along one of the common
directions. This configuration can be obtained from a solution of
supergravity as described in appendix A.
Let us wrap the in
, the fundamental string () in
and put a wave along .
The coordinates , , are assumed to be periodic,
each with radius . The metric, the dilaton
and the antisymmetric tensor are given by

(8) | |||||

and

(9) | |||

where the various harmonic function are given by (A), with the identifications and . One may also express the “magnetic” brane in terms of the dual “electric” field,

(10) |

The constant parts of the and were chosen (using a constant gauge transformation) such that the antisymmetric tensors are regular at the horizon.

Dimensionally reducing in , one gets a non-extremal black hole, whose metric in the Einstein frame is given by

(11) |

where

(12) |

This black hole is charged with respect to the Kaluza-Klein gauge fields originating from the antisymmetric tensor fields and the metric. When all charges are set equal to zero one obtains the Schwarzchild black hole. The metric (11) has an outer horizon at and an inner horizon at . The Bekenstein–Hawking entropy may easily be calculated to be

(13) |

where is the volume of the unit 3-sphere and and are Newton’s constant in five and ten dimensions, respectively.

We shall now show that this black hole is U-dual to the configuration
of the non-extremal BTZ black hole times a 3-sphere.
This will be achieved by using the shift transformation and
a series of dualities. Since neither dualities^{6}^{6}6
For T-dualities, this has been shown in [28]. S-duality
leaves the Einstein metric invariant and, therefore, it does not change the
area either. nor the shift transformation change the area of the horizon,
the Bekenstein–Hawking
entropy of the resulting solution is the same as the one of the
black hole we started from. The idea is to
dualize the fundamental string and the into a wave,
apply the shift transformation (4) and then
return to the original configuration. One sequence of dualities that
achieves that is, first, to perform
( denotes T-duality along the -direction,^{7}^{7}7
T-duality interchanges the type-IIA and type-IIB string theories.
When restricted to the fields in the NS–NS sector, the T-duality
rules are the same as those of Buscher [31]. For the R–R background
fields the corresponding rules can be found in [32].
and is the
S-duality transformation of the type-IIB string theory).
Then, the becomes a -brane, the wave a -brane and
the fundamental string a wave. So, we can use the shift transformation
(4) in to change the harmonic function ,
as in (7). In addition, the radius of is now equal
to . Next, we perform . This
yields a wave in , a in and a in
. Now, we use the shift transformation
(4) in to change the harmonic function .
The radius of also changes to .
Finally, we return to the original configuration with the inverse dualities
(no shift transformations). The final result is given
by the metric in (8), but with^{8}^{8}8It
is possible to obtain (14) and (15) below in a
single step, by combining all preceding transformations into one
element of the U-duality group.
The same comment
applies for the similar considerations in subsection 2.2.
Notice that the coordinate transformation (4) and the
subsequent duality, combine into a single
T-duality transformation along an isometry which is
space-like everywhere, but at spatial infinity, where it becomes null.

(14) |

and, in addition,

(15) |

Notice that the parameters and associated to the charges
of the original fundamental string and the solitonic five-brane
appear only in the compactification radii of and respectively,
and not on the background fields themselves.^{9}^{9}9In the extremal limit
where , , and
, with the corresponding charges kept fixed, we have to
perform the contraction
, , () and
. Then (8) and (9)
have well-defined limits, and similarly for (16) and (19)
below.

Dimensionally reducing along we find

(16) |

where

(17) |

is the metric of the non-extremal BTZ black hole in a space with cosmological constant , with inner horizon at and outer horizon at . The mass and the angular momentum the BTZ black hole are equal to and . In terms of the original variables:

(18) |

In addition,

(19) |

where is the volume form element of the unit 3-sphere. Therefore, the metric (16) describes a space that is a product of a 3-sphere of radius and of a non-extremal BTZ black hole. Notice that the BTZ and the sphere part are completely decoupled. Also all fields have their canonical value, so that both are separately exact classical solutions of string theory, i.e. there is an exact CFT associated to each of them. For the BTZ black hole the CFT corresponds to an orbifold of the WZW model based on [15, 16], whereas for and the associated antisymmetric tensor with field strength , given in (19), the appropriate CFT description is in terms of the WZW model.

We can now calculate the entropy of the resulting black hole. The area of the horizon is equal to

(20) |

whereas Newton’s constant is given by

(21) |

It follows that equals (13), i.e. the Bekenstein–Hawking entropy of the final configuration is equal to the one of the original black hole. Notice that the Newton constant in (21) contains the parameter , i.e. carries information on the charge of the original five-brane.

### 2.2 black holes

Consider the solution of type-IIA supergravity that describes a non-extremal intersection of a brane in , a brane in , a solitonic five-brane in with a wave along . The eleven-dimensional origin of this solution is described in appendix A. The coordinates , are assumed to be periodic, each with radius . The metric, the dilaton and the antisymmetric tensors are given by

and

(23) |

where the various harmonic functions are given in (A) of appendix A (but renamed as , and ).

Upon dimensional reduction in , one obtains
a charged non-extremal black hole^{10}^{10}10Extremal black hole
solutions embedded in eleven-dimensional supergravity where constructed
in [33].
In particular, these authors showed that a configuration of three intersecting
five-branes with a wave along a common string and another
one of two membranes and two five-branes reduce, upon compactification to four
dimensions, to the extremal limit of (24).
with metric in the Einstein frame
given by

(24) |

where

(25) |

The antisymmetric tensor fields and the off-diagonal part of the metric give rise to gauge fields under which this solution is charged. The usual Schwarzchild black hole is obtained by setting all charges equal to zero. The metric (24) has an outer horizon at and an inner horizon at . The Bekenstein–Hawking entropy may easily be calculated to be

(26) |

where is the volume of the unit 2-sphere and is Newton’s constant in four dimensions.

In order to show that this black hole is dual to the BTZ one, we use the same strategy as before. We dualize the solution in such a way that each brane becomes a wave, then we apply the shift transformation, and we finally return to the original configuration with the inverse dualities. For instance, the chain of dualities converts the into a wave. In order to convert into a wave one may use the dualities (starting from the original configuration) . Finally, the may be converted to by . Then, one may use the same dualities as in the previous case. The combined effect of these dualities is to change the radius of to , the radius of to , the harmonic functions to

(27) |

and the fields to

(28) |

Similarly to the
case of subsection 2.1, the parameters , and
associated with the charges of the original , and respectively,
appear only in the compactification radii of and , but not in
the background fields themselves.^{11}^{11}11
In the extremal limit we have to perform a contraction similar to the one
described in footnote 9.

After dimensional reduction in , one gets

(29) |

where

(30) |

In addition,

(31) |

where represents the field strengths and is the volume form element of the unit 2-sphere. As in the case of the five-dimensional black hole we also see that the BTZ black hole and the 2-sphere decouple completely. We also note that the second term in (29), representing the 2-sphere with the associated gauge field we have mentioned, corresponds to the monopole CFT of [34]. Equivalently, it can also be viewed as a dimensionally reduced WZW model along one of the Euler angles parametrizing the group element.

## 3 Counting microscopic states

In this section we briefly review Carlip’s derivation of the
Bekenstein–Hawking entropy formula for the BTZ black hole.
The basic idea is that
only quantum states leaving on the horizon of the black hole are relevant
in such computation, whereas those in the bulk are irrelevant.
Since the horizon represents the end of the world for an outside
observer, it is treated as a surface boundary.
This is in principle applicable in any number of dimensions, and
the problem is to be able to separate the boundary from the bulk degrees
of freedom and subsequently to quantize them.
This is a formidable task by itself
in more than three space-time dimensions, and we know of no solution to date.
However, in dimensions the
problem is trivially solved since there are no
bulk degrees of freedom at all. Moreover, as we have seen in section 2
this is enough
for our purposes, since we have mapped the problem of counting microscopic
states for the and black holes into the corresponding problem for the
BTZ black hole. The topological character of -dimensional
gravity is manifest in its Chern–Simons formulation [17, 18].
In the presence of a non-vanishing cosmological constant the action can be
written as^{12}^{12}12
We only give the bosonic part. The full supersymmetric version
has also a Chern–Simons form, but in superspace [17].
In principle, one should also keep the fermions in the derivation
of the boundary action. The latter, however,
at least in the limit of small cosmological constant,
have subleading contribution to the entropy. Nevertheless, it will be
useful to repeat the computation by including the fermions as well.

(32) |

where^{13}^{13}13Our normalizations are compatible with the representation
, ,
for the generators, and is the matrix trace.

(33) |

represents the Chern–Simons actions for some manifold and similarly for . The gauge connections are in the Lie algebra of and are given in terms of the spin connection and triad 1-forms as

(34) |

where .
We will denote
and similarly for
.
The constants and are related by .^{14}^{14}14
Notice that, since the Newton constant depends on various charges
(as follows from our discussion in the previous section), so
does . This resonates with the idea of the string-tension renormalization
employed in [35].
It is well known that,
if the manifold has no boundary, a Chern–Simons theory in
dimensions is a topological field theory.
However, if there is a non-trivial boundary ,
then the variational problem of pure Chern–Simons is not well defined unless
we specify the boundary conditions and add a boundary-action term .
This is the case of interest to us, where the non-trivial
boundary will be identified with the (apparent) horizon of the
-dimensional BTZ black hole. This, in turn, is a guideline for fixing
the appropriate boundary conditions. We will briefly repeat the arguments of
[13] (see also [36]
for a systematic general discussion of boundary
conditions and edge states in gravity).
We change coordinates from , where and
are light-cone coordinates (the precise relation can be read off by
comparing (17) and eq. (3.1) of the first article in
[13]).
Consider the boundary, with the topology of a cylinder, parametrized
by the angular variable and the non-compact variable .
Keeping , ,
, as well as their tilded counterparts,
fixed in the boundary, requires that the
action be given by
’s replaced by ’s. The total action is given by the sum of
(32) and and, as a result, the variational problem is
now well defined.
The relevant degrees of freedom in
the boundary are isolated by parametrizing ,
where is a fixed gauge connection in the boundary, and similarly
for . Then,
quite generally, it can be shown that the relevant induced action in the
boundary is the sum of two WZW actions for with opposite levels:
minus a similar term with

(35) |

As we have already mentioned, since the horizon of the BTZ black hole
at
(which in the new coordinates is located at )
is a null surface, we should demand that
the boundary be a null surface as well.
It was shown in [13] that the appropriate
boundary conditions that achieve this are
.
Moreover, we should demand that the circumference of the boundary
be the same as that of the BTZ black hole,
namely .^{15}^{15}15The actual circumference
is , where or
depending on whether the BTZ black hole
corresponds to the or to the black hole. Rescaling
, , and
effectively sets . Since, in three dimensions,
the Newton constant is accordingly rescaled.
It is possible to perform the counting of states using the original
parameters, but we prefer the rescaled ones,
thus keeping contact with the original computation by Carlip.
Then, a natural boundary
condition, which is also in agreement with the metric (17),
is .
What remains is to choose a
boundary condition for .
As there is no physical principle that has not been met
at this point, we leave its boundary value undetermined for the moment.
The aforementioned
boundary conditions are not invariant under the full two-dimensional
group of diffeomorphisms but only under rigid translations of the
angular variable . Hence, we must impose on the Hilbert space of
(35) the constraint

(36) |

where and are the zero modes of the Virasoro generators corresponding to the affine algebras for and in (35). The expectation value of , in a Hilbert space state of total level , assumes the form

(37) |

with the Casimir operators given by

(38) |

where , , are the zero modes in a Fourier series expansion of the gauge connection , i.e. and obey the Lie algebra . A similar expression holds for as well. Also and label the representation of . Recall that, we have imposed on the boundary that . Hence, the Casimir operators in (38) are positive-definite. As we shall see, this fact and simple thermodynamical considerations, allow only for the principal series representation. Using the boundary condition and the definition (34) we may express the zero modes as

(39) |

where denotes the zero mode of and encodes the remaining freedom in choosing boundary conditions. Then, using (37), we find that

(40) |

In the thermodynamic limit the configurations with maximum number of states dominate. Hence, we should maximize with respect to . It can be easily shown that the maximum value of is reached for and that it is given by

(41) |

Finally, the entropy is computed by using the fact that for a CFT with central charge the number of states behaves asymptotically at large levels as [37]

(42) |

Using the leading order in value for the central charge, i.e. , one computes the entropy to be [13]

(43) |

This is precisely the Bekenstein–Hawking entropy formula for the BTZ black hole.

We next prove that, due to boundary conditions, only principal series representations of are allowed in the thermodynamic limit, in which . This limit is what one intuitively expects from a physical point of view, but it can also be established by requiring that for the statistical description to be valid the condition should be fulfilled [38], where is the temperature of the black hole. In our case we have explicitly

(44) |

which implies that . Due to boundary conditions, (38) reduces to , where is given by (39) (computed for ). This algebraic equation is solved for to give

(45) |

where we have used (41). It is clear that the discrete series representations for which and (or if we consider the universal cover of ) and the supplementary series for which , , are not allowed if , since then becomes complex. However, this is precisely what is needed for the continuous series representation to be allowed, since in this case , . Identifying the latter expression with the one in (45), after it is rewritten so that it is valid for large , we obtain

(46) |

For the corresponding is given by an expression similar to (46), but with replaced by . Clearly, the right-hand side of (46) is positive for a sufficiently large number of states , i.e. the principal series representation is allowed.

Our final comment concerns the issue of unitarity in WZW models based on non-compact groups. In general, this is still an unsolved problem (for earlier work on the subject, see [39, 40, 41]), but in the case of the WZW model it has been argued that a consistent, unitary theory, can be obtained by restricting to highest-weight states belonging to the principal series representation [42]. In this case the current algebra character formula is the same as that of a theory of three free bosons [40]. However, the construction of modular invariants is subtle, essentially because states in the Verma module corresponding to the principal series representation do not form a closed set under the fusion rules [41]. In addition, if we try to construct modular invariants by using only principal series representations, we would need to obtain the appropriate measure of integration over all . Notice, however, that since the boundary conditions break the two-dimensional diffeomorphisms it may not be necessary to have a modular invariant formulation; only the norm of the microstates is required to be positive-definite. It would be important to reexamine these and related issues in view of the great relevance of the WZW model in black-hole physics we have uncovered.

## 4 Connection with D-branes

Since we want to compare our counting of microscopic black-hole states with the counting using D-branes, let us consider the extremal case where the latter is valid. In the D-brane picture, one constructs a configuration of D-branes that carries the same quantum numbers as the corresponding black hole. Counting the degeneracy of this configuration yields the number of microstates. When we uplift it to M-theory it becomes an intersection of membranes , five-branes and plane-wave solutions.

The effect of the shift transformation on the M-branes and on intersections of them has been studied in [24]. The result is that certain branes and intersections thereof are mapped into spaces that are locally isometric to spaces of the form , where denotes the -dimensional anti-de Sitter space, denotes the -dimensional Euclidean space and is the -dimensional sphere. We tabulate these results below. We also give the result for the -brane. Similar results hold for the rest of the branes, but only when they are expressed in the “dual -frame”, i.e. the metric in which the curvature and the -form field strength appear in the action with the same power of the dilaton [43]. In all cases, in order to arrive at the dual configuration one needs a number of compact isometries. This yields the space indicated in the second column of the table with some global identifications. For instance, the appearing below is more properly viewed as an extremal BTZ black hole (with only if a plane wave is added to the corresponding configuration in the left column).

Table 1

It is rather remarkable that these considerations distinguish branes and intersections that we already know to play a distinguished rôle for other reasons. For instance, from these configurations (with the addition of a wave in some cases) one can obtain black-hole solutions in upon dimensional reduction.

Since after the duality the asymptotic geometry has changed,
the degrees of freedom should organize themselves into
representations of the appropriate anti-de Sitter group.
The latter has some representations, the so-called singleton
representations, that have no Poincaré analogue.^{16}^{16}16These
representations, for the case of , were
discovered by Dirac [44] and named singletons by
C. Fronsdal [45].
They have appeared in studies of spontaneous
compactifications of eleven-dimensional supergravity
on spheres. In particular, the fields of the
supersingleton representation appear as coefficients in the
harmonic expansion of the eleven-dimensional fields
on the corresponding sphere. A crucial property is that the singleton
multiplets can be gauged away everywhere,
except in the boundary of the anti-de Sitter space [46].
In particular, it has been argued in the past that
the singleton representations of , , and
correspond to membranes [47], five-branes [48, 49], self-dual
threebranes [48, 49] and strings [50], respectively.
It has actually been shown that, in all cases, the world-volume
fields of the corresponding -brane form a supersingleton multiplet.
We, therefore, conclude that the membrane
, the five-brane , the self-dual
threebrane , as well as strings,
are U-dual to supersingletons.
Looking back to Table 1, we see that the anti-de Sitter
spaces appearing there, are precisely the ones we just discussed,
with one exception, the space. The boundary of is simply
a point. Thus, one deals with quantum mechanics instead of quantum
field theory. It is very tantalizing to identify the theory on the
boundary with branes. This might yield a connection with
M(atrix) theory. However, the solution factorizes
as only in the “dual-8 frame”. So, it is not
clear whether or not such an identification is correct.

What is important is that, precisely as in our discussion of the counting of states in section 3, would-be gauge degrees of freedom become dynamical at the boundary. Let us consider, for concreteness, the case of the extremal black hole. The M-theory configuration is the intersection of an wrapped in , an wrapped in with a wave along . The black hole arises after a dimensional reduction along . Let us first consider the effect of the shift transformation to each brane separately (i.e. consider a configuration with only that brane). The becomes the singleton representation of . The anti-de Sitter space has coordinates . The coordinate used to be the radius of the transverse space. The five-brane is represented by gauge degrees of freedom everywhere except at the boundary. Studying the five-brane dynamics is equivalent to studying the supersingleton dynamics of . In a similar fashion, the membrane becomes the singleton representation of (with coordinates ). After superposition the effects of the two branes cancel each other in the relative transverse directions. We end up with , where the part is along the common world-volume directions. It follows that the latter contains gauge degrees of freedom that become dynamical at the boundary. These correspond to the singleton representation of , which can be interpreted as a string [50]. Thus, we find a string living on the world-volume of the five-brane [51]. Notice that the anti-de Sitter group coincides with the conformal group in one dimension lower. Therefore, one ends up with a conformal field theory on the boundary. Since we are considering extremal black holes, the theory at the boundary is also supersymmetric. After the addition of the wave along , the becomes a massive extremal BTZ black hole. These are precisely the degrees of freedom we have counted in section 3. A similar interpretation holds also for the black hole.

Notice that the non-extremal black holes result from the non-extremal intersection of extremal branes and not from the intersection of non-extremal branes. In other words, they can be viewed as non-extremal “bound-state” configurations [29]. This means that one still has the interpretation of each brane as a singleton representation of the corresponding anti-de Sitter group. Therefore, the above discussion still applies.

## 5 Higher-dimensional black holes and further comments

Let us briefly discuss higher-dimensional
() black holes.
These cases are more complicated, since they are not connected
to three-dimensional black holes. A direct proof that the BTZ black
hole cannot appear in U-dual configurations of these black holes
is given in appendix B. Already from the discussion of the
previous section, however, it follows that the higher than five-dimensional
black holes are associated with higher than three-dimensional theories.
The black holes can be obtained from the non-extremal intersection
of with a wave, black holes from the intersection
of with a wave, and black holes from the intersection
of with a wave [29]. Hence, these black holes^{17}^{17}17
The black hole does not seem to be on an equal footing with
the rest. One may obtain black holes from
a configuration of an brane with a wave that has
an extra isometry along which one may dimensionally reduce.
This implies, however, that the corresponding sphere does not decouple.
are associated with the first three entries of Table 1.
It follows that in order to understand them one would need
to understand the boundary field theories of , and ,
respectively. Our considerations also imply that the metrics (supplied
with the appropriate antisymmetric tensor fields),
after we remove the part corresponding to the sphere,
describe solutions of gauged
supergravities in four, five and seven dimensions.
Presumably, they are black-holes solutions,
but this question deserves further study.

The fourth and fifth entries of Table 1, when supplemented with waves, correspond to the