The AdS 3 /CF T 2 correspondence in black hole physics

Transcription

1 Utrecht University - Department of Physics and Astronomy The AdS 3 /CF T 2 correspondence in black hole physics Stefanos Katmadas Master Thesis under the supervision of Prof. dr. B. de Wit June 2007

2 Abstract In the present thesis, the approach of [6] in computing the entropy of black holes in string theory is reviewed. The importance of the near horizon AdS 3 geometry and of the associated Chern-Simons supergravity is explained, followed by an exposition of the mechanism through which a Chern-Simons theory in AdS 3 induces a CFT on its boundary. Finally, the entropy in the boundary CFT is identified with the entropy found by counting microscopic degrees of freedom through the AdS/CFT correspondence. This formalism is applied to a number of examples in four and five dimensions. Full agreement with both the microscopic and macroscopic computations is established. i

3 Acknowledgements At this point, I would like to thank the people who helped in the course of writing this thesis. First and foremost, I have to thank my advisor, Prof. dr. Bernard de Wit. I am sure that his insightful and encouraging comments will be valuable to me in the future. Furthermore, I would like to thank my family and close friends for their constant support throughout the course of my studies in Utrecht (despite the few thousands of kilometers that separated me from most of them). Finally, I thank my flatmate Nikos for employing his artistic skills in the preparation of the thesis talk. ii

4 Contents 1 Introduction 1 2 D-branes and the AdS/CFT correspondence Branes in string/m-theory AdS/CFT correspondence Gauge fields at the boundary of AdS Pure Chern-Simons theories and the boundary Virasoro algebra Definitions and gauge fixing Charges at the boundary as a Sugawara construction Application to 2+1 dimensional gravity Relation with gravitational anomalies Locally AdS 3 geometries from modular transformations AdS 3 and BTZ solutions Quotients of AdS 3 and the SL(2, Z) family of solutions The partition function of the gravity theory and black hole entropy 33 7 Black holes constructed from D-branes Generalities The D1-D5 system and five dimensional black holes Wrapped M5 branes and four dimensional black holes Black rings in five dimensions AdS/CFT for black holes D1-D5 system Physics in the decoupling limit Compactification on S Relation with anomalies M5 brane on a Calabi Yau manifold Compactification on S Black rings Discussion and outlook 71 A The decoupling limit 74 B Formulae used in the S 3 reduction 75 iii

5 1 Introduction The subject of black holes has been around for almost a century in different manifestations. Their status has fluctuated over the years, from being viewed as singular solutions of the Einstein equations to becoming a major branch of research in classical General Relativity. This led to a golden age, during which the famous uniqueness theorems were proven and Hawking radiation [1] was discovered as evidence of a thermodynamic origin of the entropy formally assigned to black holes. This line of research continued to produce interesting results well into the 90 s, when Wald [2] introduced a generalised notion of black hole entropy as the Noether charge arising from the symmetry generated by the horizontal Killing vector field of the horizon. This definition is valid in any gravity theory and more importantly to the case of a higher derivative theory. Even though this picture is gratifying in the classical regime, it poses a number of highly nontrivial questions to a potential quantum theory of gravity. Putting aside issues like smoothing out the classical singularities and providing a quantum picture of the local structure of spacetime, the very existence of a macroscopic black hole entropy calls for a microscopic explanation in terms of (classically unobservable) fundamental degrees of freedom. Finding such a set of fundamental degrees of freedom and showing that they indeed lead to the known macroscopic entropy has proven to be a difficult task, which is by definition closely related to a full nonpertubative definition of a consistent theory of quantum gravity. As is well known, a leading candidate for such a theory is superstring theory, which is automatically a consistent quantum theory of gravity. In fact, it is the only framework which has allowed for a precise comparison between microscopic and macroscopic entropy. The microscopic degrees of freedom in this case come from D-branes, truly nonpertubative objects defined as endsurfaces of open strings. Their most important properties are given in a lightning review in subsection 2.1, including the supergravity solutions describing macroscopically observable superpositions of them, called p-branes. If one assumes that the observed mass and charge of a black hole is a result of a large number of D-branes wrapped on the compact directions, the p-branes can be used as building blocks that can be combined to produce a black hole. This is a valid solution of the corresponding low energy supergravity in the noncompact dimensions and a macroscopic entropy is assigned to it though the area law (or Wald s definition). As was shown in the pioneering work [40], a convenient combination of large numbers of D-branes can indeed lead to a microscopic degeneracy that leads to an entropy matching the macroscopic one. The methods of constructing black hole solutions by combining different kinds of D-branes and the ideas behind the match between the microscopic and macroscopic entropy are explained in section 7. Despite this remarkable match, the main shortcoming of the framework set up in this way is its restriction to rather special black holes, namely extremal solutions that can moreover support at least one Killing spinor. This is because the microscopic ingredients used in the modeling of black holes are themselves extremal and supersymmetric and more importantly, because the connection between the microscopic and macroscopic entropies needs the explicit assumption that the final object preserves some of the initial supersymmetry. It follows that, 1

6 with the current methods, the only familiar case of a black hole that can be treated strictly is the extremal Reissner-Nordstrøm solution when embedded in an N 2 supergravity theory, since it is known to support one Killing spinor. This has been extended to near extremal cases as well, but a full treatment of nonextremal black holes is still missing. At any rate, the fact that at least some black holes can be treated in the string theory framework is encouraging, as it shows that this approach contains the right degrees of freedom. As is evident, this microscopic/macroscopic matching of entropies is a complicated construction that depends on many issues, some of which are still under investigation. In this thesis the focus will be on the macroscopic side and more precisely on the computation of the macroscopic entropy of the black hole. At first sight this might seem easy, given the fact that the entropy is defined as the horizon area of the black hole or, more generally, through Wald s conserved charges. This is true if one has the explicit solution at hand as in the case of General Relativity, but finding and classifying all possible supersymmetric configurations in an extended supergravity and assigning to them the correct microscopic charges is far from an easy task. Moreover, this becomes much more complicated when one tries to include higher derivative corrections to the two derivative action. This is necessary for making the precise match with microscopic string counting, because the low energy effective action of string theory contains such corrections. This means that a consistent and above all supersymmetric way of including these corrections has to be found, so that the resulting black hole solutions automatically include them. This was achieved in [11] for a certain class of corrections to the leading order N = 2 supergravity arising as the low energy effective theory of Type II string theories on Calabi-Yau (CY) manifolds. The end result was found to be in agreement with the corresponding microscopic counting of [9]. In [6], an alternative way of computing the macroscopic entropy was proposed, which is quite different from the ones based on the horizon area. Its main feature is the explicit use of the higher dimensional setting and of the AdS/CFT correspondence, unlike the purely low dimensional treatments, such as in [11]. In this case, one starts from the full ten (or eleven) dimensional solution that describes the intersecting branes and considers an appropriate limit, called the decoupling limit, which zooms in the near horizon geometry. This leads to a factorised geometry with one factor always being an AdS space and the rest are compact manifolds. As briefly reviewed in subsection 2.2, the supergravity theory that results on this geometry is conjectured to be dual to the microscopic theory on the worldvolume of the branes [27]. Since the microscopic entropy stems from this worldvolume theory, it is natural to try to compute it from this near horizon AdS supergravity. Alternatively, such a computation can be also seen as a test of the AdS/CFT duality, because a possible mismatch of the entropy would invalidate it. As will be seen, in all cases of black holes in four and five dimensions where a precise microscopic derivation of the entropy is available, the worldvolume theory can be accurately approximated by a dimensional CFT. Consequently, the decoupling limit of the corresponding supergravity solutions involves similar spaces, namely an AdS 3 space times a sphere. This case will be the subject of the present thesis. In particular, we will review the approach introduced in the series of papers [6], [13], [5] in computing the entropy through the dual of 2

7 the AdS 3 space. As this involves two largely independent tasks, namely reducing the higher dimensional theories on the AdS 3 space and subsequently dealing with the resulting three dimensional theory and its AdS/CFT dual, the following sections also fall into two distinct parts. We now turn to an overview of the contents of these two parts, excluding the next section which contains basic background concepts used throughout the text. The first part is comprised by sections 3 to 6 and contains the relevant points in the three dimensional setting. First, in section 3, the Chern-Simons terms are argued to be the only relevant terms for an AdS/CFT duality computation, on the basis of the asymptotic boundary conditions imposed on the gauge fields. Using this, the boundary currents corresponding to the local symmetries of the bulk theory are found through a practical approach. This is reinforced in section 4, where a Hamiltonian formulation of the pure Chern-Simons terms is used to rederive the boundary currents in a more controlled way. In particular, the dual theory is shown to contain an affine algebra of the currents derived and an associated Virasoro algebra with a central charge equal to the Brown-Henneaux one [19]. Finally, taking advantage of the observation that three dimensional AdS (super)gravity can be viewed as a Chern-Simons theory of an appropriate (super)group, it is argued that this construction can produce the full (super)conformal algebra under which the dual theory is invariant. Then, in section 5, a small digression on the subject of the solutions of three dimensional gravity with a negative cosmological constant is made. This proves to be useful in the following discussions, as all these solutions can be uniquely described as different quotients of AdS 3 because of the peculiar nature of three dimensional gravity 1. Finally, in section 6 all the previous ingredients are put together into a derivation of the entropy. Just like in the microscopic theory, it arises as the degeneracy of the eigenvalues of the L 0, L0 operators in the boundary CFT at high temperature, but here all quantities are given in terms of the dual supergravity theory. The result is essentially the Cardy formula, as expected. The second part deals with particular examples of black holes in string theory compactifications. First, in section 7 a small introduction to the methods used to built black hole solutions from supergravity p-branes is given. After describing the general ideas, explicit constructions of black holes in four and five dimensions are discussed. These are treated one by one in detail in section 8, by considering the dimensional reduction to the near horizon AdS 3 space and finding all the relevant Chern-Simons terms. Then, a straightforward application of the results of the first part gives the entropy of the black holes in perfect agreement with the microscopic and previously known macroscopic results. The final section is devoted to a discussion of the results, more recent research and future directions. As a final comment, note that the approach of [6] was not the first time that the near horizon AdS 3 geometry has attracted attention. Initially, a purely three dimensional approach tried to use the particular simplicity of pure gravity in that dimension to quantize it by treating its boundary degrees of freedom quantum mechanically (see [49] for a review). This was later connected to higher dimensional black holes through their near horizon geometries [48], or even string dualities in simple cases [47]. This program represents another line of thought 1 In three dimensions gravity does not have any local degrees of freedom 3

8 with its own subtleties, the main one being the appearance of a hard to quantize SL(2, R) WZW model on the AdS 3 boundary. These issues will not concern us here, as the point of interest will be the AdS/CFT dual of the AdS 3 theory, which is well understood. 2 D-branes and the AdS/CFT correspondence In this introductory section, we discuss the basic aspects of the most important objects and concepts which form the basis of all descriptions of black holes in the string theory framework. This includes first of all the D-branes, the microscopic ingredients that carry the mass and the charges of the black holes. Their presence also gives rise to the degrees of freedom responsible for the macroscopic black hole entropy. We will therefore begin with a quick review of their properties and description in both the pertubative string and supergravity regimes. By employing a certain limit that concentrates on the D-brane worldvolume, this will naturally lead us to the famous AdS/CFT correspondence, the main tool in all the developments presented here. 2.1 Branes in string/m-theory For more than twenty years after the birth of string theory as a potential theory of gravity, it was thought that the only objects present in the theory were the fundamental strings used in its pertubative definition. Surprisingly, this turned out not to be true nonpertubatively, as in the early nineties it became evident that objects with various worldvolume dimensions existed in all known string theories. These cannot be seen from a string worldsheet perspective (at least not without an external hint), so that a qualitative language will be used to introduce them. The presence of extra objects in string theory can be argued for heuristically using the fact that all string theories contain antisymmetric tensor gauge fields which do not couple to anything at the pertubative level. In fact, the only exception is the NS two form present in all string theories, which couples to the fundamental string. If one requires the presence of sources for the other tensor fields as well, the possibility of the existence of extra extended objects arises. If these objects are assumed to be fundamental, their coupling should be of the form A, where W is the worldvolume of such an object and A is a tensor gauge field, as W for the familiar case of a point particle coupling to an ordinary gauge field. This has two very important implications. First, in analogy with the case a point particle coupling to a vector field, these objects must have worldvolume dimensions equal to the number of indices of the tensor gauge fields. The other one is that they should come in pairs of electric and magnetic ones, as in general one should also add magnetic sources for the gauge fields. Drawing an analogy with the four dimensional case, where the magnetic (electric) currents are the sources for the (dual) field strength, it follows that for each object with p + 1 worldvolume dimensions acting as a source for a 8 q form field strength, there must be a magnetic source for the Hodge dual p + 2 form field strength with 7 p worldvolume dimensions. 4

9 Independently of these heuristic arguments, the initial discovery that such multidimensional objects must be present came through the study of the string theory spectrum when compactified on a circle. Considering such a compactification of a closed string theory produces a spectrum of states with contributions both from pertubative excitations and winding of the strings along the circle. This spectrum is invariant under the inversion of the radius of the circle in appropriate units, an operation known as T-duality. When open strings are included, the requirement of preservation of T-duality leads to the presence of the so-called D- branes, spatially extended objects of any dimension on which open strings can end. It turned out that in any given string theory D-branes can only be stable if their dimensionality is the one dictated by the corresponding tensor gauge fields present in that theory. Moreover, it was shown that each of these D-branes carries one unit of charge with respect to these gauge fields. It then follows that they are exactly the fundamental objects described above. From now on we will call them D-branes or Dp branes, using p to denote the number of spatial dimensions of the brane. Then e.g. a D2 brane will have three worldvolume dimensions and will couple to a 3-form gauge field (only present in Type IIA string theory) and its magnetic dual will be the D4 brane as follows from the above. Finally, the magnetic object that couples to the NS two form and is dual to the fundamental string has also been shown to be present in the theory. According to the above, it has six worldvolume dimensions and is called the NS5 brane, as it arises in the NS sector that is common to all string theories. This object will not enter any discussion in the following. By the property of being the end point of open strings, the presence of a single D-brane must break at least half of the supersymmetry of the original supergravity theory, since there must be a boundary condition on its volume that relates the right and left moving spinors on the strings. As it turns out, the D-branes as defined above actually preserve exactly half of the original supersymmetries, being 1/2 BPS states with mass equal to their RR charge: M = Q. Generically, the presence of a Dp brane along the p directions imposes a projection condition on the spinor parameters of the supersymmetry transformation: ɛ R = Γ 0 Γ 1... Γ p ɛ L, (2.1) where ɛ L, ɛ R are the two chiral Majorana-Weyl spinors in ten dimensions arising from the left and right movers on the closed string. In analogy with fundamental strings, the description of D-branes can be given with the use of worldvolume actions coupled to background bulk fields, which in the low energy limit are described by a ten dimensional supergravity. The leading order woldvolume physics on the D-brane will naturally be described by a theory of point particles describing the effective interactions of the end points of the open strings. The above supersymmetry constraint is a very strong requirement for it, since in any dimension there is essentially a unique theory of point particles invariant under 16 supersymmetries, namely the maximal super Yang-Mills theory of that dimension. Starting with the extreme case of a spacetime filling D9 brane, one can see that labeling the endpoints of the strings according to the D-brane they belong to amounts to having Chan-Paton factors. Thus, the relevant gauge group is U(N) for N coincident branes, as the gauge fields mediating the interactions of the endpoints must have 5

10 two indices running up to N. It follows that the worldvolume theory is the ten dimensional super Yang-Mills theory with a new interpretation for the U(N) Chan-Paton factors. When this is extended to lower dimensional branes, one encounters the same situation of open strings but with some of the boundary conditions for their endpoints changed to the Dirichlet kind. This means that the endpoints are only allowed to vary along the directions of the D-branes present and so are the fields describing their interactions. By ignoring the normal directions, the relevant worldvolume theory in leading order is the dimensional reduction of ten dimensional super Yang-Mills theory. Except for the gauge field, the bosonic sector of this theory contains (9 p)n 2 scalars representing the transverse fluctuations of the branes. Therefore, one can describe the separation of the branes by giving different expectation values to these scalars, breaking the U(N) symmetry down to smaller groups. Restricting attention to the above worldvolume picture, the open strings can be viewed as Wilson lines connecting different charged point particles and can in fact be reconstructed as gauge theory solitonic spikes that stick out of the D-brane. Thus, the gauge excitations on the D-brane correspond to a gas of open strings having their end points on it. In the case of interest in this thesis, namely black hole constructions, these excitations will ultimately be identified as the carriers of the microscopic degrees of freedom giving rise to the entropy of the black holes constructed from D-branes. More precisely, they will be the excitations carrying momentum along specific directions. We now shift gears and briefly discuss the long range fields excited by various Dp-branes, as seen from a bulk supergravity point of view. These are the classical backgrounds that result from the presence of a large number of coincident D-branes. In the context of Type II supergravity they are called p-brane solutions (p here denotes again the number of spatial dimensions) and will later become the building blocks of the various black hole solutions of classical supergravity. In the p-brane solutions, only the metric G, the dilaton φ and the corresponding RR p + 1-form A 01...p are turned on, the value of p distinguishing between Type IIA and Type IIB supergravity (even for IIA and odd for IIB). The form of the action is: I = 1 d 10 x [ ] G e 2φ (R + 4( µ φ) 2 1 ) 16πG 10 2(p + 2)! da (p+1) 2. (2.2) The extremal form of the p-brane solutions solving the resulting equations of motion is: ds 2 = H 1/2 p ds 2 (R 1,p ) + H 1/2 p ds 2 (R 9 p ) (2.3) e 2φ = H (3 p)/2 p (2.4) A 01...p = H 1 p 1, (2.5) for integer p and H p a harmonic function on R 9 p. This solution has Poincaré invariance in p + 1 dimensions and carries RR charge. Moreover, if the harmonic function is chosen to be: H p = 1 + Q p, (2.6) r7 p the solution is asymptotically flat and can be seen to satisfy the BPS bound M = Q p, where M is the ADM mass. All these properties make these solutions good candidates to describe 6

11 stacks of D-branes. In the cases considered here, the harmonic function will always be chosen as above. The quantity Q p is the corresponding RR charge and depends linearly on the number N p of D-branes that make up this configuration as: Q p = n p g s l 7 p s N p, n p = (2 π) 5 p Γ ( 7 p 2 ). (2.7) Here, we have introduced the string coupling g s and the string length l s = α, where α is the string tension. The precise normalisation is derived by requiring that the masses expected from independent microscopic considerations [38]: M p V p = N p g s (2π) p l p+1 s match with the mass of the above solution as an asymptotic charge:, (2.8) lim g 00r 7 p = 7 p r 0 8 Q p = 16πG 10 M p, (2.9) 8Ω 8 p V p where Ω n = 2π n+1 2 /Γ( n+1 ) is the volume of the n-sphere. The ten dimensional Newton constant 2 is G 10 = 8π 6 gsα 2 4. Note that the mass is computed with respect to the transverse directions, since the p-branes are assumed to extend to infinity. By use of string dualities on toroidal compactifications, any D-brane can be transformed to a wave along a compact direction. Moreover, as mentioned above, the presence of momentum excitations along some direction is a crucial ingredient in the construction of black hole solutions. Since these charges are seen on an equal footing from the lower dimensional point of view, we give the supergravity solution describing the long range fields produced by momentum propagating on a string along a compact spatial direction. In this case, only the metric is nontrivial: ds 2 = dt 2 + dx 2 + Q K r (dx 6 dt)2 + ds 2 (R 8 ). (2.10) Again, the constant Q K is linearly related to the number of momentum excitations N along the string as Q K = 2 5 π 2 α gsn/r 2 2, (2.11) where R is the length of the compact dimension. This result for the charge can be obtained by dualising the above expression for the D-branes [38]. All the solutions above can be generalized to the nonextremal case as well [39]. For the metric, this amounts to two operations. First, the dt 2 and dr 2 terms are modified as: dt 2 f(r)dt 2, dr 2 f(r) 1 dr 2, f = 1 µ. (2.12) r7 p Furthermore, the harmonic functions associated to D-branes are changed as: H p = 1 + Q p r 7 p H p = 1 + Q p r 7 p tanha p (2.13) 7

12 in all cases. This same change is also enforced on the dilaton. On the other hand, the p-form potentials change differently for each value of p. For the D1 brane, which will be used later, the replacement to the two-form potential is: H 1 H 1 1 = 1 Q 1 r 6 H 1 1 (2.14) (H 1 here is the redefined one). The D5 brane potential does not change form (of course, now the function H 5 is the new one in (2.13)). Note that in the nonextremal case the charges shown here are defined as: Q i = µ sinh a i cosh a i, (2.15) so that only one extra nonextremality parameter µ is introduced, but we keep the charges for clarity. Finally, the metric for nonextremal momentum excitations along a string can be found by implementing a Lorentz boost along the nonextremal string to add momentum charge on it: dt d t = dt cosh a P dx sinh a P, dx d x = dx cosh a P dt sinh a P. (2.16) Then, the momentum charge takes the form (2.15) as well. All these changes consistently reduce to the extremal case when the limit µ 0, a i is taken with the charges (2.15) kept fixed including the pure momentum case. We now turn to a brief discussion on the range of parameters in which the two pictures of D-branes presented here best describe the system in question. For the description of a D-brane as an end surface of open strings to be valid, the quantum corrections to this picture must be small. This means that one must consider the loop expansion parameter on the worldvolume theory and make sure that it is small. Since for a large number N of D-branes we are dealing with a gauge theory in the large N limit, only the planar diagrams of the theory survive and the relevant loop parameter can be taken to be gy 2 M N, where g Y M is the coupling of the gauge theory. The factor of N comes in from the sum over the indices of the fields in the fundamental representation going around in the loops. As the gauge coupling is constrained to be equal to the open string coupling, which in turn is the square root of the closed string coupling g s, it follows that gy 2 M = g s. Thus, from a string perspective the hyperplane description of D-branes is valid when g s N << 1. On the other hand, the supergravity description is valid when the scale of the curvature R is much larger than the string length. The curvature scale is set by the constant Q p in (2.6), which in turn is explicitly given in (2.7). One then arrives at the following requirement on the Ricci scalar: R 7 p /l 7 p s Q p /l 7 p s = g s N >> 1 (2.17) for the supergravity picture to be relevant, which is the other extreme of the coupling parameter as compared with the previous case. What should be emphasized in both cases is that the string coupling must be kept small so that string loop corrections are negligible. It follows that the supergravity description is relevant only when the charges are very large, so that the effective coupling g s N can be large even with small string coupling. This constraint on the charges is in line with the point of view that the p-brane solutions are classical fields arising from a superposition of large numbers of microscopic states. 8

13 All the above statements about D-branes have immediate analogs in the theory that is conjectured to be the unified setting for all string theories, namely M-theory. It is known that its low energy limit is eleven dimensional supergravity, whose field content is very simple: the graviton, the gravitino and a three form gauge field. According to the discussion above, there should be an electric and a magnetic brane coupling to the gauge field with three and six worldvolume dimensions, respectively. They are named M 2 and M 5 branes, in line with the notation used for the D-branes. The low energy theories on their worldvolumes are again invariant under 16 supercharges. In particular, the worldvolume theory on the M 5 brane is invariant under (2, 0) supersymmetry 2 and includes a two form gauge field, whose excitations can be viewed as M2 branes ending on it, similar to the situation for D-branes and open strings. This last observation will be explicitly used in the following. By considering the compactification of this theory on a circle, one can find the corresponding D-branes of type IIA string theory. The M2 brane gives rise to the D2 brane or the fundamental string if it is transversal to the circle or wrapped on it, respectively. Similarly, the M5 brane becomes either the NS5 brane or the D4 brane. Note that the objects found by reducing the two electric-magnetic dual branes are electric-magnetic duals of each other, as expected. The eleven dimensional supergravity 1/2 BPS solution describing N coincident M 5 branes is: ds 2 = f 1/3 ds 2 (R 1,5 ) + f ( ) 2/3 dr 2 + r 2 dω 2 4 F (4) = da (3) = df, f = 1 + πnl3 p, (2.18) r 3 where the Hodge dual is in the five dimensional transverse space and l p is the eleven dimensional Planck length. The normalisation of the charges shown is derived in a similar way as for the D-branes. On the other hand, the corresponding solution for the M2 brane is: ds 2 = f 2/3 ds 2 (R 1,2 ) + f 1/3 ( dr 2 + r 2 dω 2 7 A 012 = f 1 1, f = π2 Nlp 6. (2.19) r 6 As there is no analog of the string coupling in eleven dimensions, the validity of the supergravity limit is controlled only by the charges, which should be large for the exact same reasons discussed for the D-branes in string theory. The two dual descriptions of D-branes and M-branes presented here show the intimate connection between geometry and gauge theory in string theory backgrounds. What is more, they provide highly nontrivial and physically interesting examples of the very different descriptions a system may have in the weak and strong coupling limits. This will become manifest when taking the so called decoupling limit, which has the property of isolating the two dual descriptions of the D-branes from their surroundings. 2 Recall that in six dimensions spinors are chiral, so that supersymmetry parameters are classified in this fashion ) 9

14 2.2 AdS/CFT correspondence The two descriptions of D-branes in the two extreme limits of the effective coupling naturally led to the idea that if one could somehow concentrate on the worldvolume of the branes on the small coupling side, a relation with the intrinsic properties of the massive object at the center of the p-brane geometry could be found. This was achieved in Maldacena s remarkable paper [26], through a procedure called the decoupling limit. As elaborated in the following, the resulting geometry on the supergravity side is a locally AdS space near the source at the center of the geometry. In view of the identification of the objects responsible for the physics at the two sides, a duality between supergravity theories on AdS spaces and the worldvolume field theories arises as a strong and very interesting possibility. Here, an elementary review of the arguments that led to the original AdS 5 /CF T 4 conjecture for a set of stacked D3 branes will be given. The core of these ideas will come back again and again in the following as a reccuring theme in different contexts. Consider the small coupling description of N coincident D3 branes as hyperplanes sitting in flat ten dimensional space. As discussed above, the low energy physics of this configuration is described by a supergravity theory on a flat background in the transverse directions coupled to the U(N) maximally supersymmetric gauge theory on the worldvolume of the branes. It turns out that the interactions between the branes and the bulk supergravity and the higher derivative corrections of the worldvolume theory are all proportional to the combination g s α and the string tension α respectively. Moreover, the gravitational coupling of the gravity theory is again proportional to g s α. If one considers the limit α 0 with r/α kept constant, where r is the radial distance between the branes, two interesting things happen. First, all the interactions of the branes with the bulk and the higher derivative terms drop out and the bulk supergravity theory is reduced to a theory of zero coupling. Thus, the system effectively decouples in two independent subsystems, namely a supergravity theory in the bulk and a pure N = 4 super Yang-Mills theory on the four dimensional worldvolume, which is known to be superconformal. The second effect of this particular limit is that it keeps the worldvolume physics intact, as the constancy of the ratio r/α makes the zoom on the D-branes possible without sending the distances on the branes to zero. In the strong coupling case, the same system is described by a geometry of the form in (2.3), carrying N units of the four form gauge field and preserving 16 supersymmetries as well. The corresponding limit α 0 now involves the steepening of the gravitational barrier between the near horizon limit and the asymptotically flat region, as the typical length of variation of the curvature goes to zero. Upon taking this limit with U = r/α fixed, the metric in (2.3) for p = 3 becomes that of the near horizon geometry, namely AdS 5 S 5 : 1 α ds2 = U 2 Q 3 ds 2 (R 1,p ) + Q 3 du 2 U 2 + Q 3 dω 2 (S 5 ), (2.20) where Q 3 = Q 3 /α is a quantity independent of α. One can show that the limit taken keeps the energies of the near horizon supergravity excitations finite in units of l s, so that they are accurately described by an AdS supergravity. Since this near horizon geometry is 10

15 separated from the asymptotically flat region by an infinite gravitational barrier as α 0, no near horizon excitation can escape to the asymptotically flat region. Conversely, as the the length scale α goes to zero, all the supergravity excitations in the flat region have very large wavelengths compared to the size of the curved region near the center and the cross section of interacting with it goes to zero. We thus see again that the physics in the limit α 0 is described by two decoupled systems: one in the near horizon region and another in the flat region. Now, since in both the strong and weak coupling cases a supergravity theory on a flat background appears, one is left with the apparent conclusion that the near horizon AdS supergravity on the strong coupling side is the dual of the weak coupling worldvolume gauge theory. Quite interestingly, all the symmetries conspire in a consistent way. In both sides we find an enhancement of supersymmetry, as the AdS 5 compactification supports twice the amount of supersymmetry the initial p-brane solution did and the limiting superconformal theory on the D3 branes requires twice as much supercharges as the original nonconformal one. Thus, this so called decoupling limit results to a doubling of supersymmetries in both sides, in a very different way. Actually, not just the number of supersymmetries, but all the various (super)symmetries of the two backgrounds match. Namely, the SO(2, 6) isometry group of AdS 5 is identified with the conformal group in dimensions. Furthermore, the SO(6) = SU(4) symmetry of the AdS 5 S 5 background is matched with the SU(4) R-symmetry of the gauge theory at the boundary. In fact, one can show that this identification extends to the whole of the supergroups on the two sides. We will provide more detailed discussions of this point in the specific applications in later sections. This spacetime match led to the idea that the CFT dual to the AdS supergravity can be thought of as living at the boundary of AdS [51]. As the free parameters of the bulk theory at the boundary are its boundary conditions, the matching prescription is that the dynamical variables of the dual CFT are the source currents of the various bulk fields at the boundary. More precisely, if O, φ are used to denote a generic operator in the CFT and its coupling respectively, a relation of the form exp φ 0 O CF T ] = Z str [φ = φ 0 (2.21) is proposed ( denotes the conformal boundary of the AdS spacetime). This states that the mean value of the exponential of any operator O is related to the partition function of the string theory on the AdS space with the boundary condition for a bulk field φ given by the value of the coupling of O in the CFT. Then, the dual bulk field φ is reconstructed through its equations of motion by this boundary condition. It follows that the expectation value of any operator in the CFT can be thought of as giving rise to a boundary current for a corresponding dual field in the bulk. Conversely, the expectation value of an operator in the CFT can be found by varying the bulk action with respect to its dual field and throwing away the contribution from the equations of motion, i.e. keeping only the boundary currents. In particular, it turns out that the given prescription implies that the metric in the bulk is the field dual to the CFT stress tensor, whereas the bulk 11

16 gauge fields are dual to conserved currents 3 in the CFT. This is exploited in section 3 in a concrete example to find the expectation values of these operators by variation of the AdS supergravity action with respect to the boundary values of the fields. Note that the crucial assumption used in writing down a relation such as (2.21) is that the supergravity background takes the form of a local product of an AdS 5 S 5 only asymptotically near the boundary, allowing for a spacetime match with the CFT to be possible. The geometry deep in the bulk can be of any form allowed by the equations of motion, as only the boundary conditions on the fields are constrained. This convenient property will be much used later. The above arguments make plausible the possibility that the gravity theory is truly dual to the boundary CFT, giving effectively a holographic character to gravity. The most remarkable property of this duality is its strong/weak coupling character, which makes it both extremely interesting and useful and difficult to prove. In particular, it obscures the exact extent of this duality, as in principle the gravity dual of the N = 4 super Yang-Mills theory could be the full string theory or just its low energy limit, the supergravity theory. The strongest version of the conjecture argued for in this section is that the full string theory on AdS 5 S 5 is dual to N = 4 super Yang-Mills in dimensions and is known as the AdS/CFT correspondence. Its current status, after almost a decade of research, is that of a conjecture that has passed a great number of nontrivial tests, to the point that it is generally believed to be true (see [27] for a classic review). This is the point of view adopted and used here as well, but in a lower dimensional setting. In the cases treated here, the systems of D-branes arise through the microscopic description of black holes in the framework of string theory. When the decoupling limit of the corresponding supergravity solutions is considered a locally but not globally AdS 3 space arises, as there will be extra parameters in the original solutions that will change the form of the near horizon region. This means that we will need a definition of the decoupling limit more precise than the one sketched above, which is given in Appendix A, where a simple set of working rules is provided. The importance of this particular decoupling limit over the simple near horizon limit 4 in the context of black hole physics is that the final theory is really dual to the stringy microscopic CFT from which the entropy stems. As will be seen in the following, this provides both conceptual and practical tools in computations. 3 Gauge fields at the boundary of AdS 3 In a number of string theory constructions of black holes, a black string solution in five or six dimensions is encountered, which upon dimensional reduction on a circle along the string yields a black hole solution in four or five dimensions. In all of these cases, the black string near horizon geometry found by employing the decoupling limit is of the form AdS 3 S p 3 Note that gauge symmetries in the bulk transform into global symmetries at the boundary, so that the conserved currents are related to these global symmetries. 4 Actually, taking the near horizon limit blindly can even change the signature of the metric in the nonextremal cases, which is surely unacceptable. 12

17 [3] [4]. By reducing to the AdS 3 space, one can study the system through an appropriate three dimensional supergravity coupled to the Kaluza-Klein modes from the sphere reduction. These three dimensional theories are related by the AdS/CFT correspondence to conformal field theories in two dimensions as explained in the previous section. Motivated by this, we study the properties of gauge fields near the boundary of asymptotically AdS 3 spacetimes. In particular, we consider the case of AdS 3 keeping the periodic identification of time 5 because the application in mind is the study of the thermodynamic properties of black holes, in which case the analytic continuation to Euclidean time will be implemented. Note that the boundary of AdS 3 is a two torus with coordinates t (the compactified time) and φ (the standard angular coordinate). Through the relation (2.21) one can compute physically interesting quantities in the CFT at the AdS 3 boundary by variation of the supergravity action as explained in connection to that equation. We initially follow [5] through the variation of the supergravity action to find the stress tensor and the currents in the boundary CFT through this prescription. Consider a single gauge field in a curved three dimensional space M with a negative cosmological constant. From the higher dimensional point of view, it can be one of the U(1) black hole charges or a nonabelian gauge field coming from the Kaluza-Klein reduction on the sphere. In the cases of interest the spheres are two and three dimensional, leading to the nonabelian gauge groups SO(3) SU(2) and SO(4) SU(2) R SU(2) L, so that it suffices to consider just SU(2). The generic form of the action including a Chern-Simons term reads: I = 1 d 3 x G (R 2l ) 1 [ T r F F + k (A da + 23 )] 16πG 2 N 4 2π A A A + I bndy, (3.1) where the gauge coupling constant g is normalised to one and can be reintroduced by adding an overall factor of g 2 to the gauge part of the action. The motivation for including a Chern- Simons term is that it may be present when reducing to AdS 3 higher dimensional theories. It is a well known result that the constant k associated with it has to be an integer for the action to be invariant under gauge transformations that are nontrivial only in the bulk, the so called proper ones. Even though we will be primarily interested in more general gauge transformations that extend to the boundary, it will be convenient to assume that k is an integer, so that all proper gauge transformations can be suppressed. As a final comment on the bulk part of the action, note that for nonabelian gauge fields there must be an extra factor of two in front of the Chern-Simons term, which is suppressed here for convenience and will be accounted for at the end of this section in the final result (3.12). The last term in the action stands for boundary terms for the gravitational and the gauge fields. The ones needed for the gravitational part are the Gibbons-Hawking term and a term associated with the cosmological constant for an asymptotically locally AdS 3 space as this is the type of space we will assume to be working in [14]: 5 What is usually done is to take the covering space of AdS 3, so that time is decompactified. Here, we keep time periodic and implicitly allow it to have an arbitrary period. Therefore, this is a generalization of the algebraic definition through which the time and the angular coordinate are constrained to have the same periodicity. See subsection 5.1 for a related discussion. 13

18 I G bndy = 1 8πG N M d 2 x ( g g ij K ij 1 ). (3.2) l Here, K ij = 1 2 ηg ij is the extrinsic curvature of the boundary in a (Gaussian normal) system of coordinates in which we can write the metric as ds 2 = G µν x µ x ν = dη 2 + g ij (η, x)dx i dx j. The coordinate η grows asymptotically as η/l log r/l, where r is the usual radial coordinate on AdS 3. The second term in this boundary integral prevents the variation of the action on the boundary from diverging. For the gauge part we will set [15]: I Y M bndy = k 8π M d 2 x gg ij T r [A i A j ]. (3.3) Assuming that k > 0, the condition that A w is fixed at the boundary must be imposed, where w = φ t, w = φ + t. This boundary condition is chosen because after varying the full action one is left in the end with the variation of A w at the boundary, so that it must be set to zero. We will comment on the case k < 0 below Eq. (3.11). A point that has to be emphasized in connection with (3.3) is that the appearance of the boundary metric is very helpful notationally but not altogether correct. This term actually depends only on the gauge fields and the modular parameter of the torus at the boundary of AdS 3. This is due to the fact that one has to introduce explicit coordinates in order to consider the fields at the boundary and therefore a specific structure of the boundary torus. The boundary term can be written in this form only when the metric of the boundary torus is assumed to be of the form ds 2 = dwd w. See [15] for the explicit calculations. In any case, this form will prove itself to be convenient when computing the induced stress tensor (as long it is not used in a careless way). These boundary terms will be justified when going to the Hamiltonian formulation in the next section, at least for the gauge field for which the construction will be completely explicit. Now, let us see how this theory looks like near the boundary of the asymptotically AdS 3 space. The overall goal will be to decompose the bulk fields near the boundary into radial and transverse parts and to analyse their leading order radial behaviour for large distances. By varying the supergravity action with respect to the transverse parts according to (2.21), the stress-energy tensor and the conserved current of the boundary CFT will be found. Ultimately, the central charge of the dual CFT will be connected to the algebra of these charges. To accomplish this, a Fefferman-Graham expansion [16] will be used for the fields near the AdS 3 boundary: g ij (η, x) = e 2η/l g (0) ij (x)+g(2) ij (x)+o(e 2η/l ), A i (η, x) = A (0) i (x)+e 2η/l A (2) i (x)+o(e 4η/l ), (3.4) along with a choice of gauge A η = 0, where g (0) ij is the flat metric on the boundary torus (ultimately this corresponds to a choice of conformal structure). This expansion expresses the requirement that the space be asymptotically AdS 3. In this three dimensional setting the first equation is just a rewriting of the boundary conditions, which is basically the statement that the (t, φ) metric grows as r 2 away from the center, just like for the AdS 3 space as can be 14

19 seen from its metric: ) ds 2 = (1 + r2 dt 2 + dr2 l r2 l 2 + r 2 dφ 2. (3.5) The second one is the simple requirement that the gauge field should approach a pure gauge radially independent matrix at large distances from any source. There are two very important implications of this expansion. The first is that the zeroth order connection must be flat. This is immediately obvious from the fact that the field strength should behave as 1/r 2 e 2η/l at large distances, combined with the given η dependence through the exponentials and can easily be verified directly. In fact, it includes possible backreactions of the metric since the given expansion depends only on the boundary conditions of the problem, namely that the metric grows asymptotically as r 2 (for it to be asymptotically AdS 3 ) and the gauge field strength falls as 1/r 2, which must be true for the exact solution. This is crucial for the discussions below, as this result holds even if higher derivative terms are added for the gauge fields (any term involving the field strength will vanish asymptotically). Moreover, it should be emphacised that the statement that the gauge connections are pure gauge at infinity is not as trivial as it sounds, because they are nevertheless allowed to be nonzero at the boundary, so that (strictly) they cannot be removed by a gauge transformation. This will become evident in the next section, where the gauge transformations extending to the boundary will be shown to correspond not to true gauge invariances but to a global symmetry of the theory. The second implication of (3.4) is that g (0) ij acts as the metric on the two dimensional boundary and the exponential in front as a conformal factor. Therefore, when one varies the action with respect to the scale of the boundary metric, it means that the radius at which the boundary is located will be varied as well. This will induce a conformal anomaly on the boundary theory (which we have not yet shown is conformal). These results also hold if higher derivative terms are added for the gravitational field as well, since AdS 3 will still be a maximally symmetric solution (but with a modified scale l), as long as the theory admits asymptotically AdS 3 boundary conditions. Now, it is straightforward to calculate the induced boundary stress-energy tensor by the prescription: T ij 2 δi =, (3.6) g (0) where the various fields are considered to be satisfying their equations of motion. For the gravitational part this requirement simplifies the computation very much, since the variation of the bulk term plus the part of the variation of the Hawking-Gibbons term proportional to η δg ij is to be set to zero. The only terms that survive by construction are the terms proportional to δg ij at the boundary, which is held fixed by the boundary conditions. The explicit expression is: δi G = 1 2 δg (0) ij ( 1 g K ij g ij K ij g ij 1 ) δg ij + bulk terms (3.7) 8πG l By the AdS/CFT correspondence prescription (2.21), the coefficient of the variation of the boundary metric is the stress tensor of the dual CFT. By substituting the expansion (3.4) and 15