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Utrecht University - Department of Physics and Astronomy The AdS3/CF T2 correspondence in black hole physics Stefanos Katmadas Master Thesis under the supervision of Prof. dr. B. de Wit June 2007 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 AdS3 geometry and of the associated Chern-Simons supergravity is explained, followed by an exposition of the mechanism through which a Chern-Simons theory in AdS3 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 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 Contents 1 Introduction 1 2 D-branes and the AdS/CFT correspondence 4 2.1 Branes in string/M-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 AdS/CFT correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Gauge fields at the boundary of AdS3 12 4 Pure Chern-Simons theories and the boundary Virasoro algebra 4.1 Definitions and gauge fixing . . . . . . . . . . . . . . . . . . . . . . . 4.2 Charges at the boundary as a Sugawara construction . . . . . . 4.3 Application to 2+1 dimensional gravity . . . . . . . . . . . . . . . . 4.4 Relation with gravitational anomalies . . . . . . . . . . . . . . . . . 18 18 21 23 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Locally AdS3 geometries from modular transformations 27 5.1 AdS3 and BTZ solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 5.2 Quotients of AdS3 and the SL(2, Z) family of solutions . . . . . . . . . . . . . . . . . 29 6 The partition function of the gravity theory and black hole entropy 33 7 Black holes constructed from D-branes 7.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The D1-D5 system and five dimensional black holes . . 7.3 Wrapped M 5 branes and four dimensional black holes 7.4 Black rings in five dimensions . . . . . . . . . . . . . . . . . . . . 37 37 42 46 49 . . . . . . . 52 52 52 55 61 64 66 70 8 AdS/CFT for black holes 8.1 D1-D5 system . . . . . . . . . . . . . . . . 8.1.1 Physics in the decoupling limit 8.1.2 Compactification on S 3 . . . . . 8.1.3 Relation with anomalies . . . . 8.2 M 5 brane on a Calabi Yau manifold . . 8.2.1 Compactification on S 2 . . . . . 8.2.2 Black rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Discussion and outlook 71 A The decoupling limit 74 B Formulae used in the S 3 reduction 75 iii 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 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 1 + 1 dimensional CFT. Consequently, the decoupling limit of the corresponding supergravity solutions involves similar spaces, namely an AdS3 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 the AdS3 space. As this involves two largely independent tasks, namely reducing the higher dimensional theories on the AdS3 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 AdS3 because of the peculiar nature of three dimensional gravity1 . 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 L0 , L̃0 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 AdS3 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 AdS3 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 with its own subtleties, the main one being the appearance of a hard to quantize SL(2, R) WZW model on the AdS3 boundary. These issues will not concern us here, as the point of interest will be the AdS/CFT dual of the AdS3 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 R form W A, where W is the worldvolume of such an object and A is a tensor gauge field, as 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 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 Dbranes, 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 N S5 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 01 . . . p directions imposes a projection condition on the spinor parameters of the supersymmetry transformation: R = Γ0 Γ1 . . . Γp L , (2.1) where L , 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 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 A01...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: Z √ 1 1 10 −2φ 2 2 I=− d x −G e (R + 4(∂µ φ) ) − |dA(p+1) | . (2.2) 16πG10 2(p + 2)! The extremal form of the p-brane solutions solving the resulting equations of motion is: −1/2 ds2 = Hp 1/2 ds2 (R1,p ) + Hp ds2 (R9−p ) (2.3) (3−p)/2 Hp (2.4) e2φ = A01...p = Hp−1 − 1, (2.5) for integer p and Hp a harmonic function on R9−p . This solution has Poincaré invariance in p + 1 dimensions and carries RR charge. Moreover, if the harmonic function is chosen to be: Hp = 1 + Qp , r7−p (2.6) the solution is asymptotically flat and can be seen to satisfy the BPS bound M = |Qp |, where M is the ADM mass. All these properties make these solutions good candidates to describe 6 stacks of D-branes. In the cases considered here, the harmonic function will always be chosen as above. The quantity Qp is the corresponding RR charge and depends linearly on the number Np of D-branes that make up this configuration as: √ 5−p 7−p 7−p np = (2 π) Γ Qp = np gs ls Np , . (2.7) 2 √ Here, we have introduced the string coupling gs and the string length ls = α0 , where α0 is the string tension. The precise normalisation is derived by requiring that the masses expected from independent microscopic considerations [38]: Np Mp = , Vp gs (2π)p lsp+1 (2.8) match with the mass of the above solution as an asymptotic charge: lim g00 r7−p = r→0 7−p 16πG10 Mp , Qp = 8 8Ω8−p Vp (2.9) n+1 where Ωn = 2π 2 /Γ( n+1 ) is the volume of the n-sphere. The ten dimensional Newton constant 2 6 2 04 is G10 = 8π gs α . 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: QK ds2 = −dt2 + dx2 + 6 (dx − dt)2 + ds2 (R8 ). (2.10) r Again, the constant QK is linearly related to the number of momentum excitations N along the string as QK = 25 π 2 α0 gs2 N/R2 , (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 dt2 and dr2 terms are modified as: dt2 → f (r)dt2 , dr2 → f (r)−1 dr2 , f =1− µ r7−p . (2.12) Furthermore, the harmonic functions associated to D-branes are changed as: Hp = 1 + Qp r7−p −→ Hp = 1 + 7 Qp tanhap r7−p (2.13) 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: H1 −→ 0 H1−1 = 1 − Q1 −1 H r6 1 (2.14) (H1 here is the redefined one). The D5 brane potential does not change form (of course, now the function H5 is the new one in (2.13)). Note that in the nonextremal case the charges shown here are defined as: Qi = µ sinh ai cosh ai , (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 → dt̃ = dt cosh aP − dx sinh aP , dx → dx̃ = dx cosh aP − dt sinh aP . (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, ai → ∞ 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 gY2 M N , where gY 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 gs , it follows that gY2 M = gs . Thus, from a string perspective the hyperplane description of D-branes is valid when gs 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 Qp in (2.6), which in turn is explicitly given in (2.7). One then arrives at the following requirement on the Ricci scalar: (2.17) R7−p /ls7−p ∼ Qp /ls7−p = gs N >> 1 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 gs 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 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 M 2 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 M 2 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 M 5 brane becomes either the N S5 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: ds2 = f −1/3 ds2 (R1,5 ) + f 2/3 dr2 + r2 dΩ24 πN lp3 , (2.18) r3 where the Hodge dual is in the five dimensional transverse space and lp 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 M 2 brane is: F(4) = dA(3) = ?df , f =1+ ds2 = f −2/3 ds2 (R1,2 ) + f 1/3 dr2 + r2 dΩ27 32π 2 N lp6 . (2.19) A012 = f − 1 , f =1+ r6 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. −1 2 Recall that in six dimensions spinors are chiral, so that supersymmetry parameters are classified in this fashion 9 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 AdS5 /CF T4 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 gs α0 and the string tension α0 respectively. Moreover, the gravitational coupling of the gravity theory is again proportional to gs α0 . If one considers the limit α0 → 0 with r/α0 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/α0 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 α0 → 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/α0 fixed, the metric in (2.3) for p = 3 becomes that of the near horizon geometry, namely AdS5 × S 5 : 1 2 U 2 2 1,p dU 2 ds = ds (R ) + Q̃ + Q̃3 dΩ2 (S 5 ), 3 α0 U2 Q̃3 (2.20) √ where Q̃3 = Q3 /α0 is a quantity independent of α0 . One can show that the limit taken keeps the energies of the near horizon supergravity excitations finite in units of ls , so that they are accurately described by an AdS supergravity. Since this near horizon geometry is 10 separated from the asymptotically flat region by an infinite gravitational barrier as α0 → 0, no near horizon excitation can escape to the asymptotically flat region. Conversely, as the the length scale α0 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 α0 → 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 AdS5 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 AdS5 is identified with the conformal group in 3 + 1 dimensions. Furthermore, the SO(6) ∼ = SU (4) symmetry of the AdS5 ×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 Z h i exp φ0 O = Zstr φ = φ0 (2.21) ∂ CF T ∂ 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 gauge fields are dual to conserved currents3 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 AdS5 ×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 AdS5 × S 5 is dual to N = 4 super Yang-Mills in 3 + 1 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 AdS3 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 limit4 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 AdS3 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 AdS3 × 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 [3] [4]. By reducing to the AdS3 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 AdS3 spacetimes. In particular, we consider the case of AdS3 keeping the periodic identification of time5 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 AdS3 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 AdS3 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: Z Z √ 1 2 k 2 1 3 I=− d x −G R − 2 − T r F ∧ ?F + A ∧ dA + A ∧ A ∧ A + Ibndy , 16πGN l 4 2π 3 (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 ChernSimons term is that it may be present when reducing to AdS3 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 AdS3 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 AdS3 , 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 G Ibndy 1 = 8πGN √ 1 ij . d x −g g Kij − l ∂M Z 2 (3.2) Here, Kij = 21 ∂η gij is the extrinsic curvature of the boundary in a (Gaussian normal) system of coordinates in which we can write the metric as ds2 = Gµν xµ xν = dη 2 + gij (η, x)dxi dxj . The coordinate η grows asymptotically as η/l ' log r/l, where r is the usual radial coordinate on AdS3 . 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]: Z k √ YM Ibndy = − (3.3) d2x gg ij T r [Ai Aj ] . 8π ∂M Assuming that k > 0, the condition that Aw̄ is fixed at the boundary must be imposed, where w = φ − t, w̄ = φ + t. This boundary condition is chosen because after varying the full action one is left in the end with the variation of Aw̄ 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 AdS3 . 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 ds2 = dwdw̄. 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 AdS3 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 AdS3 boundary: (0) (2) gij (η, x) = e2η/l gij (x) + gij (x) + O(e−2η/l ), (0) (2) Ai (η, x) = Ai (x) + e−2η/l Ai (x) + O(e−4η/l ), (3.4) (0) along with a choice of gauge Aη = 0, where gij 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 AdS3 . 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 r2 away from the center, just like for the AdS3 space as can be 14 seen from its metric: dr2 r2 ds = − 1 + 2 dt2 + + r2 dφ2 . (3.5) r2 l 1 + l2 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/r2 ∼ 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 r2 (for it to be asymptotically AdS3 ) and the gauge field strength falls as 1/r2 , 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. (0) The second implication of (3.4) is that gij 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 AdS3 will still be a maximally symmetric solution (but with a modified scale l), as long as the theory admits asymptotically AdS3 boundary conditions. Now, it is straightforward to calculate the induced boundary stress-energy tensor by the prescription: 2 δI T ij = p , (3.6) −g (0) δgij(0) 2 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 δgij is to be set to zero. The only terms that survive by construction are the terms proportional to δgij at the boundary, which is held fixed by the boundary conditions. The explicit expression is: Z 1 √ 1 1 G ij ij ij δI = − −g K − g Kij g − δgij + bulk terms (3.7) 2 ∂ 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 (0) identifying gij with the boundary metric as explained, one manipulate indices): 1 (2) k (2)i (0) (0) (0) Tij = gij − g i gij + T r Ai Aj − 8πG 8π gets the result (we use g (0) to 1 (0) k(0) (0) A A gij , 2 k (3.8) when the gauge part is added. This is trivial to compute since the Yang-Mills term does not give any boundary terms and the Chern-Simons term is topological, leaving only the boundary term (3.3) to vary. Actually, the gravitational part here is the Brown-York result [17] written in terms of the fields in the expansion above and modified by the extra boundary term. From this, it is easy to reproduce the Brown-Henneaux result [19] for the central charge of the boundary CFT. Taking the trace of (3.8) and using the following result [18] (2) g (0)ij gij = l2 (0) R , 2 (3.9) which connects g (2) to the Ricci scalar of the two dimensional boundary, we find that Tij c 3l satisfies the relation Tii = − 24 R with a central charge equal to c = 2G . Now, the currents at the boundary have to be found as well. As mentioned when introducing the boundary term for the gauge field, Aw̄ is to be held fixed at the boundary. According to (2.21), the induced boundary current is defined as the variation with respect to the boundary (0) condition for Aw̄ modulo the equations of motion: J w̄ = p 2π −g (0) δI (0) δAw̄ ⇒ J w̄ = kA(0) w . (3.10) To make the notation continuous, we use w, w̄ coordinates for the gauge part of the stressenergy tensor, which is now written as: g Tww = k (0) (0) A A . 8π w w (3.11) This is problematic for k < 0, because it is not bounded below. In order to fix this, the sign of the boundary term has to be changed. In that case, it is the variation of Aw that doesn’t cancel, so that we are forced to change our choice of gauge field components to keep fixed when varying. By changing the indices as w ↔ w̄, the current is seen to become rightmoving instead of the above leftmoving one. This may seem a bit artificial, but it will be justified in the next section where all this will be seen from a Hamiltonian point of view and the associated Virasoro and affine algebras will be constructed explicitly as the algebras of asymptotic symmetries of the solutions of the theory. From that side, the form of the boundary term for the gauge field on which both the stress tensor and the currents depend crucially will be justified as well. That however has its own drawbacks, due to the breaking of manifest Lorentz invariance inherent to the Hamiltonian formalism, that complicates the contact with a definite gravitational stress tensor in the boundary theory. This is the reason we preferred to introduce the stress tensor through this more "hands on" method. 16 In the case of interest, charged black holes, the gauge fields are abelian for the most part. However, there will always be at least one non abelian as well, coming for the Kaluza-Klein reduction as mentioned above. To be more specific, when considering the D1-D5 system on R4,1 × M4 × S 1 to get a 1+1 dimensional (4, 4) supersymmetric CFT on the circle, there is still an SO(4) ∼ = SU (2)L × SU (2)R rotational symmetry around the spatial noncompact dimensions (see section 8.1 for a discussion of the D1-D5 system). This is exactly the R-symmetry of the (4, 4) supersymmetric theory on the circle. By the AdS/CFT correspondence this is dual to the IIB supergravity description of this system, which near the horizon becomes AdS3 × S 3 × M4 , having the same SO(4) rotational symmetry. This allows us to identity the corresponding SO(4) algebras. The important point is that the supersymmetry algebra constrains the central charge of the CFT to be c = 6k, where k is the level of the SU (2) affine algebra, identified with the k above, as will be seen in the next section. The same applies for the case of M 5 and M 2 branes wrapped on a 4-cycle of a six manifold and a circle, leaving a (0, 4) supersymmetric CFT with an SO(3) ∼ = SU (2)R R-symmetry on the circle. In this case the SO(3) rotations can be identified with rotations on the sphere of the near horizon AdS3 × S 2 × M6 . To conclude this section we give the generalizations of the above formulae when multiple gauge fields are present, considering all of them to be abelian, adding only the nonabelian ones coming from the Kaluza-Klein reductions above. In is setting, the constant k becomes a matrix which we write in block diagonal form so that the positive and negative definite subspaces decouple. Assume that k IJ is a positive definite matrix (the positive definite part of the full matrix), associated with the leftmoving currents and −k̃ IJ is a negative definite matrix (the negative definite part of the matrix) giving the rightmoving ones, with I = 0, . . . to label the various fields. Note that the index I may run up to different integer values for the two sectors in general. The final expressions are: g Tww = k IJ k̃ IJ AIw AJw + ÃIw ÃJw 8π 8π (3.12) ik IJ ik̃ IJ AIw , J˜w̄ = ÃI w̄ , Jw̄ = J˜w = 0, (3.13) 2 2 where we removed the zero label for brevity and moved to the Euclidean signature for later reference. Here, the zero index is defined to correspond to the nonabelian fields. This is enough, since in the cases above there is only a leftmoving SU (2) field being nonzero for the M-brane case and both a leftmoving and a rightmoving for the D1-D5 case6 . In both cases, we use the rotational invariance on the sphere to set to zero all but the Aa and Ãa fields for (0)3 a = 3. Thus, when writing A0w above, what is really meant is Aw belonging to the SU (2)L R-symmetry and accordingly for the tilded fields. This is consistent with the extra factor of two suppressed for the nonabelian case in (3.1), as it is cancelled by the trace over the square of the matrix representation 2i σi of the generators of SU (2). Thus, the two last equations are correct for SU (2) gauge fields as well only under the assumption that the Aw ’s are actually the A3w components of the full gauge fields and this is enough for the following developments. Jw = 6 This is so because the left or right SU (2) gives rise to Chern-Simons terms with positive or negative coupling k respectively 17 4 Pure Chern-Simons theories and the boundary Virasoro algebra In this section we review the mechanism by which a theory with no physical degrees of freedom like pure Chern-Simons theory in three dimensions can imply nontrivial dynamics at the boundary of the spacetime it is defined in. This has an immediate connection with the previous section, since near the boundary of the asymptotically AdS3 space the Yang-Mills terms for the gauge fields drop out (recall that the zeroth order gauge field A(0) is a flat connection), so that the dynamics is purely of the Chern-Simons type. What is more, using the fact that the usual Einstein-Hilbert action in three dimensions can be written as a ChernSimons theory as well, it is evident that this framework provides a unified point of view for the two kinds of fields of the previous section. Therefore, we consider a single nonabelian Chern-Simons gauge field here and extend our discussion later. As will be seen, gauge transformations extending to the boundary of spacetime relate different solutions of the Chern-Simons theory, thus giving rise to nontrivial dynamics at the boundary. This boundary theory is governed by the group of asymptotic symmetries of the space of solutions of the bulk theory, that is the sum of spacetime and group symmetries. The interesting quantities to compute are the currents related to these asymptotic symmetries, for which a variation of the bulk action must be considered. Effectively, this is done by modding out the part that gives the equations of motion, since a restriction to flat gauge connections is made from the outset. This is exactly the AdS/CFT prescription in (2.21), leading to the conclusion that the currents found can in fact be identified with the currents and the stress tensor of the previous section. Ultimately, the algebras of these currents lead to an affine algebra and a Virasoro algebra coming from its Sugawara construction. This becomes more intuitive in view of the fact that there is no use of a spacetime metric and the boundary is only equipped with a conformal structure, leading to invariance under the full conformal group in two dimensions. Note finally that in this setting the form of the boundary term is not chosen from the beginning, but is explicitly constructed, so that the following discussion is selfcontained. In fact, this construction was considered in an unrelated context [20] before the AdS/CFT correspondence was put forward and provides external support to it. 4.1 Definitions and gauge fixing Consider the pure Chern-Simons theory rewritten in Hamiltonian form on a three dimensional spacetime of the form R × Σ (here we use separate space and time coordinates, as appropriate in the Hamiltonian setting, so that the notation x refers to the position on the two dimensional surface Σ): Z Z k 2 ij a b a b dt d x Kab Ȧi Aj + A0 Fij . (4.1) I= 4π Σ 18 Observe that A0 is a Lagrange multiplier7 . Here, Kab is a symmetric metric on the Lie algebra, a b c Ai Aj . We initially disregard the used to raise and lower indices and Fija = ∂i Aaj − ∂j Aai + fbc existence of a boundary, or alternatively assume all fields and gauge parameters to vanish in a well behaved manner as one approaches the boundary. Reading off the bracket between the components of the gauge field: 2π ab ij (2) Aai (x), Abj (y) = K δ (x − y), k (4.2) it is evident that the boundary conditions for the gauge field cannot be chosen arbitrarily because the conjugate momentum would be constrained as well. k ij With respect to this bracket, the constraints Ga = 4π Kab Fijb coming from the variation of the Lagrange multiplier form a closed algebra and act on the gauge field in the following way: c {Ga (x), Gb (y)} = fab Gc (x)δ (2) (x − y), Ga (x), Abi (y) = −Kab Di δ (2) (x − y), (4.3) so that they generate the gauge transformations. To get the ordinary gauge transformations in phase space, one has to contract the Lie algebra index of the generators with an arbitrary Lie algebra vector η a and integrate to find the corresponding charge: Z η a Ga . (4.4) G[η] = Σ Indeed, by a small calculation, the functional derivative of the above charge is found to be: Z Z k k δG k ij ij a δG = ηa Di δAj = − ij Di ηa δAaj ⇒ Dj ηa , (4.5) = a 2π Σ 2π Σ δAi 2π where a boundary term was thrown away. Using this, the algebra (4.3) becomes: {G[η], Aai (x)} = Di η a , {G[η], G[λ]} = G[[η, λ]], (4.6) as expected. Note that upon gauge fixing this algebra and the whole theory trivialize as a result of the above constraints that set the field strength to zero and thus force the gauge field to be pure gauge. Turning to the situation when a boundary is present, the dynamics become interesting due to the possibility of having gauge transformations which do not vanish at the boundary. This means that although the bulk dynamics continue to be trivial by the pure Chern-Simons equations of motion, there are still nontrivial dynamics at the boundary. In this case, the bracket (4.2) remains the same, but (4.5) does not: Z Z I k k k ij a ij a δG = ηa Di δAj = − Di ηa δAj + ηa δAa , (4.7) 2π Σ 2π Σ 2π ∂Σ 7 Note that there is also a difference of a boundary term proportional to the one in (3.1), which will show itself later in the analysis. 19 R ∂Σ A0 Aφ between this action and where we see the presence of a boundary term which spoils the differentiability of the charge G with respect to the gauge field. In order to fix this, a boundary term Q[η] should be included in (4.4), defined by the requirement that its variation has to be: I k δQ[η] = − ηa δAa , (4.8) 2π ∂Σ canceling the boundary term. The inclusion of this boundary term implies the need for a corresponding boundary term in the action such that its variation with respect to the Lagrange multiplier A0 leads to a charge of the form G + Q. Hence, this procedure fixes the boundary terms in the Lagrangian. By considering different cases of gauge transformations explicit expressions for Q[η] will be found later. With this definition the modified algebra is: Z δG k ij k = Dj ηa ⇒ {G[η], G[λ]} = G[[η, λ]] + ηa Dk λa dxk − Q[[η, λ]], (4.9) δAai 2π 2π ∂Σ as can be easily verified. This means that in case the last term is not zero (and as we will see it is not) for every reasonable choice of vectors, the algebra will acquire a central charge. It is then no longer possible to consistently set the modified gauge generators G[η] to zero. As a result, the states of the theory are not gauge invariant anymore and the gauge group has become a symmetry group which acts on them. In fact, after the gauge is fixed, the algebra of charges becomes: {Q[η], Q[λ]}D = Q[[η, λ]] + C[η, λ] (4.10) where this is now the Dirac bracket and we renamed the last two terms in (4.9) as C. This is a nontrivial algebra of the global charges of the states under the symmetry group. The dynamics are not trivial either, since the gauge modes have become dynamical. The action that governs them can be found by considering a gauge transformation of (4.1) and noting that it varies into a boundary term identical to the action of a chiral WZW model on the two dimensional boundary [22]. As the main focus here is on an AdS/CFT interpretation, this boundary theory will not be considered in any detail. Nevertheless, its existence is conceptually crucial, as it provides the AdS3 degrees of freedom dual to the CFT of interest. One should be carefull in distinguishing between the two CFT’s, as one of them is an induced theory equivalent to an AdS3 theory with no bulk degrees of freedom and the other is the AdS/CFT dual of that AdS3 theory. We now briefly mention the gauge fixing choices assuming that Σ has the topology of a disk, which is the case of interest for us. Referring to [20] for the actual details and the consistency check, we note that the choice of field components is simple because they are constrained to be a flat connection. From now on the choice Ar = a = const, Aφ = Aφ (φ) (4.11) will be made, which is preserved if Dr (∂φ A0 ) = 0, or equivalently, if A0 is a gauge transform of a function of φ alone (the angle on the boundary circle). In fact, Aφ can be of the form 20 b−1 A(φ)b, where b = b(r) is a group element generated by Ar = a and likewise for A0 . We disregard this for the time being, but we will have to remember to factor out this dependence when considering specific solutions (see the BTZ black hole application below). Then, all the following apply unchanged. In order to be consistent with diffeomorphism invariance, we also choose8 A0 = −ξ i Ai ξ i = (−∂φ ξ, ξ) (4.12) as a boundary condition, where ξ = ξ(φ). This particular choice of ξ i is arbitrary, but consistently leads to a subalgebra of (4.10) that will be identified as the Virasoro algebra. Nevertheless, it can be seen as the most general generator of diffeomorphisms leaving the boundary conditions invariant. In the present context, where no metric is used, there are no other boundary conditions except the conformal structure of the boundary, so that ξ is analogous to the arbitrary function solving the two dimensional conformal Killing equation and leads to the full two dimensional conformal group at the boundary. This is exactly as in the original Brown-Henneaux approach [19], where two vectors like ξ i are found (see below). As a final point, note that the above gauge fixing suggests that the radial component of the gauge field at the boundary is a gauge invariant quantity. An intuitive explanation is that if one tried to change its value through a gauge transformation, the radial derivative of a gauge group element would have to be introduced. As the boundary is at a constant radius, this is not well defined there. This invariance in fact includes even diffeomorphisms, because any diffeomorphism can be written as a gauge transformation modulo the equations of motion (see next subsection). With these gauge choices in mind, we now turn to the promised derivation of the current algebra and the Virasoro algebra at the boundary. 4.2 Charges at the boundary as a Sugawara construction At this point it is possible to arrive at the results for the boundary stress tensor and current quoted in the previous section. In order to find the currents one has to look at arbitrary gauge transformations with parameters η a , which do not vanish at the boundary. In this case the integration of (4.8) is trivial (just remove the δ-variations): Z k Q[η] = − dφ ηa Aaφ (4.13) 2π From this final form it is easy to see that the boundary term in the action that produces this boundary contribution Q when varied with respect to the Lagrange multiplier A0 is: δIbdry δQ k = = − (Aφ − A0 ) δA0 δη 4π ⇒ Ibdry = − k A0 (Aφ − A0 ) , 4π (4.14) where (4.12) was used with Ar = 0 as in the previous section and with a constant ξ = 1. This variation is exactly equation (3.10) (modulo the conventional constant). The other current 8 This can be relaxed when not considering diffeomorphisms 21 J w vanishes by (4.12). Upon inserting an appropriate integration constant proportional to A2φ R and taking into account an extra boundary term proportional to ∂Σ A0 Aφ coming from the rewriting of (4.1) as the Chern-Simons form appearing in (3.1), this matches precisely with (3.3). On the other hand, the central charge of the algebra is: Z k C[η, λ] = ηa Dk λa dxk − Q[[η, λ]] = 2π ∂Σ Z Z Z k k k a k a k ηa ∂k λ dx + ηa [Ak , λ] dx − Q[[η, λ]] = ηa ∂k λa dxk , (4.15) = 2π ∂Σ 2π ∂Σ 2π ∂Σ a nonzero result. Note that if k < 0, the same equations apply upon setting k → −k and Aφ → −Aφ , so that we get a rightmoving current as before. Finally, by Fourier decomposing Aφ the algebra (4.10) gives the associated affine algebra with C as above: ∞ Aaφ 1 X a inφ = J e k −∞ n ⇒ b Jna , Jm D c + iknK ab δn+m . = −fcab Jn+m (4.16) Note that the constant k is identified with the level of the algebra so that it is an integer, as it should if the Chern-Simons term is to be gauge invariant in the bulk. Next, we turn to the stress tensor. In order to physically justify the derivation of the stress tensor, recall the fact that in a Chern-Simons theory diffeomorphisms are contained in the gauge transformations as can be seen by looking at the Lie derivative of the gauge field: Lξ Aµ = ξ ν ∂ν Aµ + Aν ∂µ ξ ν = ξ ν (∂ν Aµ − ∂µ Aν ) + ∂µ (ξ ν Aν ) = ξ ν Fνµ + ∂µ (ξ ν Aν ) − [ξ ν Aν , Aµ ] = ξ ν Fνµ + Dµ (ξ ν Aν ) , (4.17) so that on shell this is identical to a gauge transformation. Therefore, the case when the gauge transformations are of the form η = −ξ i Ai has to be considered as well. Along with the gauge conditions (4.12) this is a proper diffeomorphism. Plugging this into (4.8) and using the fact that Ar = a = const., we integrate easily to find: Z Z k 1 φ 2 1 φ 2 k r Q[ξ] = dφ aξ Aφ + ξ Aφ + ξ a = dφξ(φ) A2φ + 2a∂φ Aφ + a2 , (4.18) 2π 2 2 4π where the explicit form of the vector ξ i was used and the last term is an integration constant chosen for later convenience. Moreover, the central charge is found by9 : Z Z Z Z k k k 1 ka2 i j r φ 2 dφ η ∂φ λ = dφ ξ Ai ∂φ ζ Aj = dφ [ξ, ζ] aAφ + [ξ, ζ] Aφ + dφ ξ r ∂φ ζ r 2π 2π 2π 2 2π Z ka2 = Q[[ξ, ζ]] + dφ ξ i ∂φ ζi , (4.19) 2π 9 The commutator [ξ, ζ] is the usual Lie bracket for ordinary vectors and we replace the covariant derivative by an ordinary one because T r([Ai , Aj ]Ak ) = 0 for i, j, k = r, φ 22 so that the last term is the central charge C of the algebra: Z ka2 {Q[ξ], Q[ζ]}D = Q[[ξ, ζ]] + dφ ξ i ∂φ ζi . 2π (4.20) Upon Fourier transforming this becomes: ∞ A2φ 1X Ln einφ + 2a∂φ Aφ + a = k −∞ 2 ⇒ {Ln , Lm }D = i(n − m)Ln+m + ic n(n2 − 1)δn+m , (4.21) 12 with c = 12ka2 . In the quantum case this manifestly becomes the Virasoro algebra, so that we find conformal invariance of the two dimensional theory at the boundary. Finally, by use of the Fourier transform for Aφ introduced above, the modes Ln can be expressed as: ∞ 1 X ka2 a b Ln = Kab Jm Jn−m + inaa Jna + δn , 2k −∞ 2 (4.22) where we recognize the Sugawara construction, as one might have anticipated by the results of this and the previous section. In fact, this is presicely the gauge part of the stress tensor of the previous section modulo a shift induced by a nonzero value of the radial component of the gauge field. For later reference, we give the algebra in the case of multiple gauge fields, following the conventions stated in connection with Eq. (3.12). In this case the Ln ’s are defined as the Fourier transform of a sum of terms like in (4.21), corresponding to the expression in (3.12). Moreover, a second set of such terms will come from the right moving Chern-Simons terms as explained above, which will be denoted by tilded quantities. Then, the full algebra is made up of the Virasoro algebra in (4.21) (along with a tilded version) plus the following ones: I [Lm , JnI ] = −nJm+n , [Jn , Jm ] = m k IJ δm+n , 2 (4.23) along with the tilded ones. Note that we changed the definition of the Jn ’s by a factor of 1/2 in order to comply with (3.12) and the literature, where Jw = 21 J w̄ is used. The term proportional to Jn+m is absent by definition for the abelian fields and vanishes for our choices given below Eq.(3.12) for the nonabelian one. The final conclusion is that a pure Chern-Simons theory on a three dimensional manifold induces a conformal theory on the boundary of that manifold. This CFT has a global symmetry group identical to the bulk gauge group, so that its operator algebra contains the affine algebra of that group and the Virasoro algebra in the form of a Sugawara construction. This can be used in a (super)gravity theory as well, as will be now reviewed. 4.3 Application to 2+1 dimensional gravity The above construction can be also applied to the case of an AdS3 gravity theory using the fact that three dimensional gravity can be written as a Chern-Simons theory with gauge group SO(2, 2), which is the group of isometries of AdS3 . Through this procedure, the well 23 known Brown-Henneaux result and the associated Virasoro algebra at the boundary can be easily derived. In fact, this construction can be generalized to a supergroup, so that any AdS3 supergravity theory can be written as a Chern-Simons theory [23]. The result is that a Virasoro algebra with the same central charge as in the pure gravity theory is induced at the boundary, irrespectively of the extra fields in the gravity supermultiplet [24]. This is not surprising, given that any such supergravity will be based on right and left supergroups realising a supersymmetrisation of the bosonic SO(2, 1), so that instead of the Virasoro algebra, an appropriate superconformal algebra will arise at the boundary. Since one should be able to consistently reduce to the original pure gravity result by keeping just the Virasoro algebras, the central charge should also be the same. Here, only the pure gravity case will be treated, corresponding to a pure Chern-Simons theory with a SO(2, 2) ∼ = SO(2, 1)+ × SO(2, 1)− gauge group. As has been shown in [25] this Chern-Simons theory is equivalent to three dimensional Einstein gravity with a negative cosmological constant. The gauge field for gravity is Aµ = eaµ Pa + ωµa Ja , where eµ and ωµ are the vielbein and spin connection and the generators Pa , Ja satisfy the SO(2, 2) algebra10 . Defining ea 1 Aa± = ω a ± , Ja± = (Ja ± lPa ) (4.24) l 2 one gets two disjoint so(2, 1) algebras for the J ± ’s. Here, l is the radius of the AdS3 . The following action is then equivalent to the Einstein-Hilbert action suplemmented with a negative cosmological constant equal to −l−2 : Z 1 1 a b a b c I = k(I+ − I− ), I± = ηab A± ∧ dA± + abc A± ∧ A± ∧ A± , (4.25) 4π 3 l is set. Note that the second part has a negative coupling constant. In this case we if k = 4G expect to get two Virasoro algebras out of this theory and that they can be identified with the rightmoving and the leftmoving central charges of the boundary CFT. To be more explicit, a direct application of the ideas in the previous subsections yields two Virasoro algebras (one for each of the ± cases): ± L± n , Lm D = i(n − m)L± n+m + ic± n(n2 − 1)δn+m , 12 (4.26) with c± = 12ka2± = 3l a2 . As there is one leftmoving and one rightmoving Chern-Simons term, G ± these Virasoro algebras and central charges are identified with the corresponding untilded and tilded ones of the previous subsection. Note that there is no SO(2, 1) affine algebra corresponding to the local Lorentz transformations because the presence of a boundary breaks the full three dimensional Lorentz invariance. It follows that only the gauge transformations (4.17), that correspond to coordinate transformations need to be considered. To complete the explicit form of the algebra, a± must be found. This is done by considering any one of the solutions in the class of interest, namely the asymptotically AdS3 ones, as the 10 We use one group index for the Lorentz generators J instead of two for convenience of the presentation. The usual form can be obtained by contracting Ja with abc 24 boundary condition for the vielbein is always the same. For later reference we choose the BTZ black hole solution [28] (see subsection 5.1 for more details). The metric takes the form ds2 = − 2 2 (r2 − r+ )(r2 − r− ) 2 r+ r− 2 l2 r2 2 2 dr + r dφ + dt + dt . 2 2 ) )(r2 − r− l2 r2 (r2 − r+ lr2 (4.27) The vielbein corresponding to this metric is : e0 = (r+ dt − r− dφ) sinh r, e1 = ldr, e2 = (−r− dt + r+ dφ) cosh r, (4.28) and the usual torsion constraint gives the spin connection as: ω0 = 1 (−r− dt + r+ dφ) sinh r, l ω 1 = 0, ω2 = 1 (r+ dt − r− dφ) cosh r. l (4.29) Then, the components of the fields along the radial and angular directions are found to be: ± A± r ≡ a± = ±J1 , A+ φ = r+ − r− + J2 , l A− φ = − r+ + r− − J2 . l (4.30) Note that we factored out the radial dependence of A± φ coming from a conjugation with ±ra± b=e as explained above. 3l a2 = 2G , which is exactly the BrownTherefore, the central charge in this case is c± = 3l G ± Henneaux result. By substituting in the expansion (4.21) for the zero index case and identifying the total energy (mass) and angular momentum of the bulk theory with the energy and momentum contained in the induced stress tensor, the mass M and angular momentum J of the solution are easily found as: L± 0 − 1 c = (lM ± J) 24 2 =⇒ M= 2 2 r+ + r− , 8Gl2 J =− r+ r− , 4Gl (4.31) These are the standard forms for the mass and angular momentum of the BTZ solution (see subsection 5.1). Note that upon setting r− = 0 one gets the AdS3 metric (3.5) out of (4.27) in the case 1 M = − 8G . In fact, the last equations show that L± 0 = 0 for AdS3 . Furthermore, noting that for both the AdS3 and the BTZ solution the stress tensor is constant because the gauge field is constant, all other L± n for n 6= 0 will vanish. We then conclude that AdS3 is the "vacuum" of the boundary CFT. When considering the supersymmetric case, it is actually found that it is the vacuum of the NS sector, whereas the BTZ black hole is viewed as the vacuum of the Ramond sector. As a final remark we point to the fact that it is not completely obvious how to make contact with the results for the gravitational part of stress tensor in the previous section. This is due to the fact that the asymptotic conditions were formulated using the metric which is quadratic in the vielbein. A further difficulty is that the dynamical field that should be relevant from the present point of view is g (2) , and not the zeroth order metric, which was used to raise and lower indices in the previous section. In any case, the stress tensor is quadratic in the vielbein -that is linear in g (2) - as expected. For a recent related discussion see [21]. 25 4.4 Relation with gravitational anomalies By the Chern-Simons reformulation of gravity in (4.25) one sees that for the standard Einstein - Hilbert action to emerge the two coupling constants must be absolutely equal: k + = k − = k. Here we consider the case when this is not so. The most important change is that now the action varies with a boundary term proportional to the difference k + − k − under gauge transformations. This can be seen from the fact that if e.g. k + > k − , the action can again be recast as the ordinary gravitational action with k = k − plus a pure Chern-Simons term for the gauge field A+ with coefficient k + −k − , which is well known to vary by a boundary term (see (4.33) below). In order to translate this back into the more familiar form used in General Relativity, a small manipulation is needed. Except for the Einstein-Hilbert term already used, the only other combination available in three dimensions is the gravitational Chern-Simons form and this is how the extra term will be viewed. Indeed, a general linear combination of the left and right Chern-Simons forms I+ , I− introduced in (4.25) can be expressed as: 1 1 k+ I+ − k− I− = (k+ + k− )(I+ − I− ) + (k+ − k− )(I+ + I− ). 2 2 (4.32) Using the usual torsion constraint to write everything with respect to the tangent bundle valued connection form Γαβµ , the first term becomes the action for AdS gravity as before. The second term and its variation under a gauge transformation δΓ = dv + [Γ, v] can be written11 as: Z Z 2 3 ICS [Γ] = β T r ΓdΓ + Γ ⇒ δICS [Γ] = β T r (vdΓ) , (4.33) 3 ∂AdS3 where β is a constant proportional to the difference k + − k − . Now, this Chern-Simons term and its boundary variation are identical to the last part of an anomaly descent chain leading to the variation under coordinate transformations of a two dimensional action suffering a gravitational anomaly [30]: Z c − c̃ δI = T r (vdΓ) . (4.34) 96π Here, c, c̃ are the left and right central charges respectively. This is very natural from the point of view we pursued in this section. Exactly because k + 6= k − , the corresponding right and left central charges don’t turn out to be equal, their difference being proportional to the difference of the coupling constants. This is interesting for calculating black hole entropy, since this can be turned around to find the difference of the central charges of the dual CFT just by looking at the coefficient in front of the gravitational Chern-Simons term that might be present. Then, if one of them is known, the other one can be found as well. This is crucial when higher derivative terms are present, as a manifestly chiral decomposition in terms of 11 To be absolutely precise, there is an extra local boundary term in this transition, which corresponds to going from a situation where local Lorentz invariance is broken at the boundary to one where diffeomorphism invariance is broken [46]. This term is nonchiral, so that it does not interfere with the argument that follows. Here we choose to break diffeomorphism invariance for ease of presentation. 26 two Chern-Simons terms is no longer possible and it becomes difficult to find the central charges through the full action using the above procedure. In the cases of the black hole constructions briefly mentioned at the end of the previous section, the underlying supersymmetry algebra gives us what we need. A very efficient way of finding the central charges is to look at the coefficient of the corresponding left and right moving Chern-Simons terms for the Kaluza-Klein gauge fields in the action. As seen above, this is equal to the level of the affine algebra at the boundary. Then, the central charges are constrained by the relation c = 6k implied by the superconformal algebra of the boundary CFT. Moreover, their difference is controlled by a possible gravitational ChernSimons term, so that this can be applied even if only one of the right or left moving sectors are supersymmetric. Therefore, both central charges can be found if the gauge and gravitational Chern-Simons terms in the action are known. These are well known in the cases we mentioned above and furthermore it is known that they do not get any higher loop corrections from string theory, so that we get the exact central charges out of this procedure. In the following we will see how they can be used to compute the entropy of a black hole with whose decoupling limit involves a locally AdS3 geometry. 5 Locally AdS3 geometries from modular transformations In this slightly disjoint section, the geometry of the solutions to the three dimensional Einstein equations with negative cosmological constant is described through SL(2, Z) quotients of global AdS3 . Though a bit technical, this discussion will be useful for later reference and in deriving the Cardy formula for the entropy of a three dimensional black hole. This in turn is crucial for later developments. 5.1 AdS3 and BTZ solutions As is well known, three dimensional gravity has no propagating degrees of freedom. This is evident from the fact that in three dimensions the Riemann tensor has only six independent components, which are linearly related to the components of the Ricci tensor: g µν Rαβγδ = gα[γ R δ]β − gβ[γ R δ]α − gα[γ g δ]β Rµν (5.1) 2 Thus, we only need the Ricci tensor to describe the geometry completely. Using the Einstein equation there are no degrees of freedom left and there are no gravitons. Any possible nontrivialities can come only from the global structure of spacetime. In the case of interest here, a negative cosmological constant is introduced, with the result that the local geometry is that of AdS3 . In fact, a naive computation can lead to the conclusion that it is the only solution to the Einstein vacuum equations (we use this example to fix notation in an obvious way): Z √ 1 2 1 I=− −g R − 2 + Ibdry ⇒ Rµν = 2 gµν . (5.2) 2π l l 27 The boundary term here is analogous to the one introduced in section 3. Along with (5.1), the Einstein equations show that the solution is a maximally symmetric space. Locally, the metric can always be cast in the form: r2 dr2 2 2 2 2 ds = − 1 + 2 dt + (5.3) 2 + r dφ . r l 1 + l2 In the case this is can be actually done globally, this is known as the AdS3 space. Its global topology is of the form R × D2 , with Dn the n-dimensional disk. Furthermore, it can be shown [27] that the conformal boundary of AdSp+2 is R × S p (with R denoting the time direction). In the p = 1 case it is a cylinder, as can easily be seen explicitly by the above metric at large radius: dt2 l2 dr2 2 2 2 ds = 2 + r − 2 + dφ , (5.4) r l which after a conformal rescaling at a constant r indeed describes R × S 1 , because of the periodicity of the angular coordinate. To be more precise, the metric (5.3) is the covering space of the more traditional definition of AdS3 through the algebraic equation: − t21 − x21 + x22 + x23 = −l2 , (5.5) which leads to a periodic time coordinate. When retaining this compactification of the time coordinate, the topology of the space is that of a Lorentzian signature torus whose interior is the bulk of AdS3 space. This is the kind of space that was considered in section 3. As our interest will be in the thermal properties of the gravitational solutions, the Euclidean case will be useful as well. The analytic continuation of this Lorentzian solid torus to imaginary time gives Euclidean thermal AdS3 , whose topology is that of a solid torus. In particular, since this torus resulted from a periodic identification of an naturally noncompact coordinate, namely time, it follows that the time direction corresponds to the noncontractible cycle of the torus12 . On the other hand, the cycle corresponding to the angular variable on the initial D2 was contractible in the bulk of AdS3 from the beginning and it should continue to be so. As we will see below, this does not hold for other solutions of the Einstein equations in general. Another solution of the these equations is the BTZ black hole [28]: 2 2 (r2 − r+ )(r2 − r− ) 2 l2 r2 r+ r− 2 2 2 dt + 2 dr + r dφ + dt . ds = − 2 2 l2 r2 (r − r+ )(r2 − r− ) lr2 2 (5.6) This geometry is locally AdS3 , the only difference being its global properties. Showing and generalizing this result will be the purpose of the next subsection. The most important of its properties is the existence of two horizons at r+ , r− (with r+ > r− ) and the total mass and angular momentum Komar integrals for this spacetime: 2 2 r+ + r− M= , 8Gl2 J =− 12 r+ r− , 4Gl (5.7) We recall at this point that on the boundary of a solid torus one can always distinguish the two cycles by the fact that one of them (namely the one with the smaller radius) can be contracted in the bulk of the torus. 28 where M l ≥ |J|. Furthermore, the temperature and entropy are equal to: −1 πr+ ∂S 2πr+ 2 2 = , T = = (r+ − r− )/2πr+ . S= 4G 2G ∂M J (5.8) Here, the entropy is found by the area law (or, in this case, length law) and the temperature by the thermodynamic relation between the entropy and the mass-energy (or by calculating the surface gravity at the horizon and dividing by 2π). The extremal case is defined as M l = |J|, or r+ = r− and this is what we will always refer to as the extremal BTZ solution in the following with no reference to electromagnetic charges. Note that the (necessarily extremal) zero mass limit is not the AdS3 vacuum as one would expect from experience with the asymptotically flat 1 case, but a rather special black hole [29]. The AdS3 vacuum arises at M = − 8G , as mentioned above. For later reference, we give the explicit expressions for the thermal geometries corresponding to the above. For AdS3 the only change is a change of sign in the dt2 term: r2 dr2 2 2 2 ds = 1 + 2 dτ 2 + (5.9) 2 + r dφ , l 1 + rl2 whereas for the BTZ case there is an extra explicit i appearing: 2 2 2 ) 2 (r2 − r+ )(r2 − r− l2 r2 ir+ r− 2 2 2 ds = dτ + 2 dr + r dφ + dτ , 2 2 l2 r2 (r − r+ )(r2 − r− ) lr2 (5.10) In order for this to be real, r− is now required to be pure imaginary so that the temperature in (5.8) is strictly nonnegative. 5.2 Quotients of AdS3 and the SL(2, Z) family of solutions The purpose of this subsection is to review the construction of a family of asymptotically AdS3 solutions of the Einstein equations with negative cosmological constant [31] [32] [5]. This family arises from all the nonequivalent boundary conditions that can be imposed so that the result is a locally AdS3 space. By the observation above that the boundary of AdS3 is a torus, a natural choice of such boundary conditions is the set of all inequivalent tori. In fact, this produces all the known locally AdS3 solutions. Recalling that AdS3 is the group manifold of SL(2, R) ∼ = SO(2, 1), which is in turn an analytic continuation of SO(3) ∼ = SU (2), we start from the group manifold of SU (2). Consider the following parametrisation of an SU (2) matrix: ! t1 + ix1 x3 − ix2 (5.11) g= ⇒ det g = 1 ⇔ t21 + x21 + x22 + x23 = 1, −x3 − ix2 t1 − ix1 where the radius l above was absorbed by rescaling. Form the last relation follows that by analytically continuing to imaginary values two of the xi ’s, AdS3 as defined through (5.5) will 29 arise, as the group will transform to SL(2, R). Doing the same for the third one too, thermal AdS3 arises. Introducing the parametrization: ! − sin ρ ei(u+ū) cos ρ eu−ū , (5.12) g= cos ρ e−u+ū sin ρ e−i(u+ū) with 0 ≤ ρ < ∞ and u ∈ C, the metric ds2 = − 21 T r[g −1 dgg −1 dg] is the following: ds2 = sin2 ρ (du + dū)2 − cos2 ρ (du − dū)2 + dρ2 . (5.13) Now, by a uniform rescaling of all the variables by i, this metric becomes that of local AdS3 : ds2 = − sinh2 ρ (du − dū)2 + cosh2 ρ (du + dū)2 + dρ2 , (5.14) if an overall minus sign is neglected. This analytic continuation has thus brought us to the analytically continued SL(2, R) group manifold. In this form, the SL(2, R) × SL(2, R) ∼ = SL(2, C) isometry group of AdS3 is manifest as can be checked by multiplying (5.12) on the right and the left by SU (2) matrices and analytically continuing. The space described by (5.14) not yet equivalent to thermal AdS3 because of the topology of the latter space. The topology of thermal AdS3 is that of a solid torus as mentioned above, whereas this is not built in the above construction. In order to get exactly thermal AdS3 , the double periodicity 2u ∼ 2u+2πi ∼ 2u+2πiτ has to be imposed, which for any given τ defines a torus on the complex plane. In fact, the fist periodicity was already there, since the coordinate u in 5.12 can only be defined modulo iπ anyway. This will prove to be very important, because the 2u ∼ 2u + 2πi cycle is always contractible on the space discussed above. This follows from the fact that it is a constant ρ cycle which can be contracted by reducing the radius ρ. The remaining periodicity can be implemented by taking the quotient of the above space by the right and left additive group: ! ! eiπτ 0 e−iπτ̄ 0 g∼ g (5.15) 0 e−iπτ 0 eiπτ̄ By this construction it is clear that τ is the modular parameter on the resulting torus if we demand that the 2u ∼ 2u + 2πiτ cycle is a fundamental one. This is needed because the above definition is insensitive to a translation by 2π which would then give a torus with modular parameter τ + 1 (a Dehn twist of the previous one). What is important in this case is that this cycle is always noncontractible because what we essentially did is to identify periodically an initially noncompact direction, namely the real part of u in these conventions. Since the same identification of a noncompact coordinate was encountered when passing to thermal AdS3 it is natural to identify the noncontractible cycle with the time direction. Now, by going back to (5.13) and choosing the coordinate u in the way suggested by the above considerations one indeed finds AdS3 . Choosing 2u = i(φ + it) one readily sees that (5.13) becomes: ds2 = sinh2 ρdφ2 + cosh2 ρdt2 + dρ2 , (5.16) 30 along with the identifications: φ ∼ φ + 2π φ ∼ φ + 2π<τ, t ∼ t + 2π=τ contractible circle - space direction noncontractible circle - time direction + twist. Note that there is an extra Dehn twist controlled by the real part of the modular parameter, <τ next to the expected imaginary time periodicity of =τ . By the change of radial coordinate sinh ρ = r, this metric becomes the one of Euclidean thermal AdS3 in (5.9). Figure 5.1: AdS3 The two distinct classes of locally AdS3 solutions. On the left (right), the boundary of global (BTZ solution) is shown, for which the angular cycle is contractible (noncontractible). Note that the global topology is the same. As one might have expected, the reason of going through all the above construction was not to get back just ordinary AdS3 . Finding AdS3 as a result is related to the specific conditions set by the boundary torus. In fact, by choosing different SL(2, Z) transformations of the modular parameter τ in (5.15) a whole family of locally AdS3 solutions is generated. This corresponds to the discrete cases of all the boundary tori that could appear in principle and will be further restricted in a moment. As the simplest nontrivial case, consider changing τ → −1/τ . In this case one finds the identifications 2u ∼ 2u + 2πi ∼ 2u − 2πi/τ , which for the choice of coordinates 2u = −i(φ + it)/τ become: u ∼ u + 2πi ⇒ φ ∼ φ − 2π<τ, t ∼ t − 2π=τ, u∼u− 2πi τ ⇒ φ ∼ φ + 2π, t ∼ t, contractible circle noncontractible circle Note that now it is the φ cycle that is noncontractible and that the modular parameter of the boundary torus is −1/τ . To show that this corresponds to a black hole, all one has to do is to substitute the above choice for u into (5.13) and change the radial coordinate as: sinh2 ρ = (r2 − =τ 2 )/|τ |2 only to get the Euclidean version of the BTZ black hole: 2 (r2 − (= τ1 )2 )(r2 + (< τ1 )2 ) dr2 dt 2 2 2 ds = U (r)dt + + r dφ + , U (r) = (5.17) U (r) (=τ <τ ) r2 r2 The temperature of this black hole is 1/T ≡ β = −2π=τ , given both by the periodicity of the time coordinate and the explicit formula (5.8) when the two horizons are at r+ = = τ1 , r− = i< τ1 . 31 The geometric difference between this solution and AdS3 with respect to the contractible and noncontractible cycles is pictorially shown in Figure 5.1. This can be generalized easily by taking the quotient in (5.13) by a general SL(2, Z) trans+b formation of the form τ → aτ and defining coordinates as 2u = cτ i+d (φ + it). Then the cτ +d +b modular parameter of the torus becomes aτ and (5.13) becomes again a solution of the cτ +d Einstein equations. Note that the above includes the case c = 0, d = 1 which gives back the initial AdS3 . In fact, the above construction gives AdS3 or (5.17) for any pair of integers c, d that are not relatively prime, because of the freedom of redefining τ → τ + 1. So, what is left is a Γ∞ /SL(2, Z) quotient. Here, Γ∞ is the subgroup of the modular group generated by τ → τ + 1. We therefore see that all the known regular and black hole geometries arising as solutions of the Einstein equations with a negative cosmological constant in three dimensions can be classified in the above scheme as different foldings of the space of 2 × 2 hermitian matrices. All of them have the same local structure as AdS3 , as expected on general grounds. In fact, they can also be seen as quotients of that space with little extra effort, starting from the above construction [28], [37]. τ > H HH α β H HH j =β 1 τ 1 Figure 5.2: Taking a quotient of (blue) one ends up with AdS3 AdS3 τ= 1 α 1 along the time direction. When the resulting period is larger than one again. When it is smaller (red), the modular parameter of the boundary torus is inverted and a noncontractible space direction appears, hence a black hole. In purely geometric (or rather pictorial) language, this process of taking the quotient of AdS3 is seen in Figure 5.2, where the simplest case of taking the quotient along the time direction is shown. The convention of keeping the modular parameter in the fundamental domain of SL(2, Z) is enforced. As the cycle with the smaller period is always the contractible one, it becomes obvious that this quotient can produce black holes in exactly the same way as above. In hintside, this structure could have been expected [31]. Using an AdS/CFT reasoning in this case, or more concretely the equivalence between AdS3 gravity and a CFT at the boundary established in subsection 4.3, it follows that equivalent states on the boundary torus should be 32 a result of equivalent geometries in the bulk. In this picture, the modular invariance on the torus should reflect on a corresponding invariance of the gravity theory solution spectrum. The only subtlety here is our "physical insistence" to keep distinct notions of space and time. This is the essential difference between a rewriting of AdS3 and a black hole, because from a purely geometric point of view we can always trivially flip the naming of the two cycles when choosing the u coordinate and get AdS3 back. This is the reason why we have full invariance under the τ → τ + 1 generator of the modular group in the final answer and not under the inversion of τ , the first one corresponding to a Dehn twist with the same contractible and noncontractible cycles and the second to a flip of them, thus giving a black hole. The sum of these two sets of solutions is then manifestly invariant under the modular transformations of the boundary torus. In any case though, we keep the time and space cycle labeling intact. 6 The partition function of the gravity theory and black hole entropy In this section the various results on locally AdS3 geometries seen until now will be used to define the entropy of AdS3 black holes using the boundary CFT. This is different from the usual (super)gravity definition of the entropy as the horizon area of the black hole, but leads to the same result. This is not surprising, as three dimensional gravity has no local degrees of freedom anyway, so that any interpretation of the entropy should be expressible with no reference to the bulk. The most important point used is that the geometry is always asymptotically locally AdS3 , so that near the boundary it can be seen as a rewriting of AdS3 space as in section 5. This means that the properties of all known locally AdS3 backgrounds can be found by applying SL(2, Z) transformations on the AdS3 solution, as explained. This allows for a semiclassical determination of the gravity partition function, which is identified with the CFT one. Then, the entropy can be found with required ingredients only the central charge and the eigenvalues of the L0 , L̃0 operators, which must be found independently for each solution. As the stress tensor can be found by examining the asymptotic properties of the solutions, the nontrivial task will be the determination of the central charge and this will be the main concern in the examples of the later sections. We define the partition function of the gravity theory Zgr in the standard way as a Euclidean path integral with an action found by Wick rotating a corresponding action on a Minkowskian signature space. The specific structure of the action will not be of much importance at this stage. All that is required from it is that it fully incorporates the local degrees of freedom of the theory in a smooth gravity approximation and that it allows for asymptotically AdS3 solutions. What we will do here is to follow [5] in finding the entropy in leading order, an approach which resembles closely the standard way of introducing the Cardy formula, which will essentially be the final result. The starting point of the discussion will be the previous result for the stress tensor and the currents of the CFT on the boundary of an asymptotically locally AdS3 geometry, applied 33 to the case it asymptotes to global AdS3 . For the gauge fields this means adding flat gauge connections near the boundary, with the boundary conditions described in section 3. If this is to be the AdS3 vacuum, the holonomy of these fields associated with the spatial circle should H vanish: Aφ dφ = 0 for any distance from the center, so we set that AIw = −AI w̄ ⇒ Aφ = 0 (6.1) for all gauge fields. This constraints all the gauge fields in this setting to be constant according to the discussion in section 4, but this will change for more general configurations. The Euclidean version of the variation of the action with respect to the boundary metric is: Z p i α i ˜α 1 ij (0) 2 (0) δI = d x −g T δgij + J δAα + J δ Ãα (6.2) 2 2π 2π In order to use this, we need a precise parametrisation of the metric variation on the boundary torus. Recall that in the discussion connected with Eq. (3.3) it was emphasized that the appearance of the whole metric is just a matter of notational convenience and that only the modular parameter of the boundary torus comes into play13 . To make this variation explicit, we introduce coordinates which both run in [ 0, 2π ): i−τ i − τ̄ w− w̄ τ − τ̄ τ − τ̄ 2 1 − iτ 1 + iτ 2 dz + dz̄ . ⇒ ds = 2 2 z= ds2 = dwdw̄ (6.3) (6.4) Now, g (0) does not depend on z and in the AdS3 case neither does Tww , as we already know from the discussion in section 4.3 that it is constant and equal to −c/48π. The same goes for the flat connections as well, but since we are integrating over the two dimensional boundary, only the zero modes matter in any case. The integration becomes trivial: Z p i w̄ i ˜w 1 ww (0) 1 w̄w̄ (0) 2 (0) δI = d w −g T δgww + T δgw̄w̄ + J δAw̄ + J δ Ãw = 2 2 2π 2π τ2 τ2 ˜ 2 4iπ −Tww δτ + Tw̄w̄ δτ̄ + Jw δAw̄ + Jw̄ δ Ãw ⇒ π π iπ (cτ − c̃τ̄ ) + πτ2 (A2w̄ + Ã2w ), (6.5) I(τ ) = 12 (0) where we changed variables only on the δgww terms, using the fact that the stress tensor and the gauge fields are constant and (3.12) was substituted. This shows that AdS3 is a finite action solution of the Euclidean theory (an instanton) with the above value of the action. The same procedure can be also followed for the black hole solutions, even if the action contains higher derivative terms. This follows from the fact that if AdS3 is a solution of 13 To be precise, the argument was about the gauge field but also holds for the gravitational field. since it can be written as a Chern-Simons theory. In particular, the actual AdS3 boundary conditions for the metric require that the boundary be a torus without any specific metric. But for the same reason as for the gauge field case, the introduction of coordinates leads to a specific modular parameter. 34 the classical equations of motion14 , the same will hold for any solution with the same local geometry modulo global identifications, such as the SL(2, Z) family of solutions discussed above. By the result for the Virasoro generators of the BTZ black hole given in section 4.3, we know that the stress tensor is again constant so that a similar result is expected. Now, given the considerations of the previous section, it is clear that the full characterization of the global geometry comes from the modular parameter of the boundary torus, so that the action of a solution in the SL(2, Z) (or Γ∞ /Γ) family of solutions is: I = I( pτ + q ), rτ + s (6.6) with I(τ ) as in (6.5). Note the digression from the standard practice of naming the four integers to avoid confusion with the central charge. It follows that in the lowest order approximation the path integral can be written as the sum over the full set of instantons found: X pτ +q Zgr = e−I( rτ +s ) (6.7) {Γ∞ /Γ} The two temperature limits of this sum give back the two extreme geometries, that of AdS3 and that of the BTZ black hole. Suppressing the gauge fields for clarity, consider the case of low temperatures, =τ = β → ∞. Then, the terms with r = 0 dominate the sum and we get that the absolute minimum is ordinary AdS3 (q = 0 and p = s = 1). In the case of high temperature, when =τ → 0, we find the opposite behaviour and we get the BTZ black hole as an absolute minimum (p = 0 and r = −q = 1). Now we make the connection with the CFT side using the AdS/CFT correspondence. That means equating the canonical partition function of the CFT to the one in (6.7) for the gravity side: i h c̃ c I I (6.8) ZCF T ≡ T r e2πiτ (L0 − 24 ) e−2πiτ̄ (L̃0 − 24 ) e2πizI q e−2πiz̃I q̃ ∼ Zgr [τ, A, Ã]. In order to make the match appropriately, so that an equality can be put in place of the equivalence, one has to specify the precise relation between the currents and gauge connections in the path integral formulation on the right hand side and the charges and chemical potentials in the trace. In order to find the normalisation, consider the normalised coupling of the gauge connection introduced in (3.10): Z 1 dwdw̄ (Aw Jw + Aw̄ Jw̄ ) . (6.9) 2π By use of the definition of the charges15 : I dw I I q =2 J , 2πi w I q̃ = −2 14 I dw I J , 2πi w̄ (6.10) We stress again that the AdS3 radius will be different from the inverse square root of the cosmological constant if higher derivative terms are included, affecting the central charge through the Brown-Henneaux result 15 The two comes from the convention in section 3 35 and computing the integrals in (6.9) as in the computation of (6.5), the result matches the one in the CFT trace if: zI = −iτ2 AI w̄ , z̃I = iτ2 ÃIw . (6.11) Finally, the relation between the Lagrangian and Hamiltonian in the two formulations must R be taken into account. This amounts to the standard shift of A2t introduced by the Legendre transformation. Using (6.11) to write it as a function of the z’s, the final relation between the path integral of the gravity theory and the partition function of the boundary CFT is: h i π(z 2 +z̃ 2 ) c c̃ iz iz̃ I I e τ2 T r e2πiτ (L0 − 24 ) e−2πiτ̄ (L̃0 − 24 ) e2πizI q e−2πiz̃I q̃ = Zgr [τ, , ] (6.12) τ2 τ2 All this being set up, the entropy of the black hole can be computed using purely thermodynamic relations. Thus, we consider the limit when only the black hole state contributes (for high temperature). This means that the limit on the CFT side in which only one state contributes as well, namely the one corresponding to the BTZ black hole, has to be considered. This implies that the eigenvalues of L0 and L̃0 should be large since τ → 0 for the BTZ black hole. Then, the trace becomes a single term, and we can identify: F π(z 2 + z̃ 2 ) π(z 2 + z̃ 2 ) c̃ 1 iπ c 2πi 2 2πi 2 − ≡ lnZgr − − z + z̃ = = −I(− ) − = − τ τ2 τ τ2 12 τ τ̄ τ τ c c̃ ) − 2πiτ̄ (L̃0 − ) + 2πizI q I − 2πiz̃I q̃ I , (6.13) 24 24 where the expression for the action of the BTZ black hole from (6.6) and the canonical formula were used. Note that here τ is viewed as temperature, since we know by the previous section that it corresponds to the period of the noncontractible cycle (and that its imaginary part turns out to be the temperature). Now, the entropy is found as the Legendre transform of the quantity −F/τ with respect to τ, τ̄ , zI , z̃I : ∂ Fτ ∂ Fτ ∂ Fτ ∂ Fτ F + τ̄ + zI + z˜I = S =− +τ τ ∂τ ∂ τ̄ ∂zI ∂ z̃I s s c q2 c̃ c̃ q̃ 2 c 2π L0 − − + 2π L̃0 − − , (6.14) 6 24 4 6 24 4 lnZCF T = 2πiτ (L0 − where we note the relations found from (6.13): L0 − 1 ∂ lnZgr c z2 c + , = =− 24 2πi ∂τ 24τ 2 τ 2 2 q I = − k IJ zJ , τ (6.15) along with the corresponding tilded ones. This is the entropy for a general black hole in AdS3 , which is essentially the Cardy formula for large L0 , L̃0 , as expected. Note however that for the charged black hole solutions considered in the following one can ignore the charges, because the contribution in L0 , L̃0 from the gauge parts of the stress tensor conveniently cancels with the explicit charges shown, as one can easily verify using (3.12) and (6.10). This is because the gauge fields are constant near the boundary, so that only zero modes contribute to L0 , L̃0 . For more general configurations, the 36 integration in (6.5) will involve a sum of opposite modes of the gauge fields (the zero modes of the A2 terms) and the full expression in (6.14) has to be used. The above result can be used in higher dimensions, based on the appearance of a factor of locally AdS3 space in the near horizon geometry of a number of string theory constructions of black holes. Thus, the argument put forward in [6], [5] is that the entropy of this three dimensional black hole should be identified with the entropy of the higher dimensional one. This makes use of the AdS/CFT correspondence, with the near horizon locally AdS3 space being dual to the 1 + 1 dimensional CFT found above. This in turn is identified with the microscopic CFT on the D-brane worldvolume. These statements will become more precise in the next sections. 7 Black holes constructed from D-branes Having set up the background material for the three dimensional theories and the CFT’s they induce on the AdS3 boundary, we now shift gears and go on with describing the higher dimensional black hole solutions and their construction. In the past decade or so, there has been great progress in the study of black holes in the framework of string theory. Here, a rather brief account of the constructions used will be given and we will frequently use results stated here in the following sections. The staring point is a general discussion of the ideas behind the string theoretic description of black holes. Then, the structures behind a number of specific systems in four and five dimensions are exposed in more detail. The unifying feature of all these systems is that their low energy limit is accurately described by a 1 + 1 dimensional CFT on an effective string in five or six dimensions. This means that, according to the AdS/CFT correspondence, their decoupling limit should lead to a locally AdS3 space times a sphere (describing the angular coordinates) and possibly some other compact manifold. This is indeed the case as will be seen in the next section, where this limit is considered and the microscopic CFT along with the entropy computed from it are matched to the CFT at the AdS3 boundary and the entropy found through the procedure in section 6. As this section is meant as a discussion of the full supergravity solutions describing the systems examined later, it is not essential to the development of the exposition of the AdS3 based approach to the computation of black hole entropy. Therefore, it can be skipped by a hasty reader, as only the final solutions will be used in later sections. 7.1 Generalities Quite generally, any attempt to produce a microscopic description of something that could be interpreted as a black hole in the noncompact directions must include two basic ingredients: a structure that appears as a pointlike source in the noncompact dimensions and the existence of an explanation of the large entropy through internal degrees of freedom. The leading approach that seems to contain the right degrees of freedom is the string theoretic one. In 37 this picture, black holes arise as a system of D-branes wrapped on internal manifolds, so that the fluctuations of the D-branes give the desired internal degrees of freedom to built the Bekenstein-Hawking entropy. This procedure has so far been successful for BPS extremal and near extremal black holes in extended supergravity, embedded in the string setting. Here we give a general discussion of the ideas involved, based on the by now classic reviews [38] and [27], as well on [39], to which we refer for more details and further references. Constructing black holes from branes The modeling of black holes in string theory rests on the following ideas. The starting point is a system of intersecting D-branes wrapped on some compact internal manifold in flat transverse space, carrying a set of p-form charges. The reason that more than one kind of branes is needed is the requirement of having an everywhere regular dilaton and nonzero horizon area of the final black hole. The kinds of D-branes and intersections is not arbitrary either and is chosen in such a way that the final state preserves some of the initial supersymmetry (see below), which means that it is a BPS state. Then, a counting of the massless modes of the effective low energy theory is performed, which corresponds to a counting of the degeneracy of the system, that is essentially the exponent of the entropy. As expected from the fact that the fluctuations of the D-branes correspond to a gas of open strings ending on them, these low energy degrees of freedom are described by (classically) massless open strings stretching between the D-branes. Moreover, the presence of more than one D-brane or, even better, different kinds of D-branes dramatically increases the number of possibilities for the positions of the open string end points. Now, if the kinds of D-branes are chosen properly and their numbers are taken to be very large, it can indeed be arranged that the degeneracy is of the order of magnitude as found in classical General Relativity. This is necessary anyway, as the macroscopic supergravity theory is only valid in this limit and so is the entropy that is derived by it. All the above correspond to a small coupling description of the system and are far from the familiar spacetime picture of say, the Schwarzchild black hole. As explained in section 2, this description is valid when gs N 1, where gs = eφ is the string coupling and N is the number of D-branes (or the typical scale of that number for the case of different kinds of branes). The black hole description in terms of a curved spacetime is found by letting the coupling become strong: 1 gs N < N , where the string coupling is still kept small (gs < 1) to avoid large string loop corrections. We are now at strong coupling and the flat space description is no longer relevant. The appropriate picture is a supergravity solution carrying the same set of charges, which is constructed by the p-brane solutions of Type II supergravity presented in section 2. As will be explicitly seen later in this section, upon dimensional reduction of the compact directions along the branes, a system of intersecting D-branes can lead to a supergravity solution describing an extremal black hole that preserves the same amount of supersymmetry and carries the same charges as the initial configuration. The above mentioned microscopic entropy is then associated to this black hole. This identification of the degeneracies in the two regimes is possible because the microscopic states in a super38 symmetric configuration satisfy a BPS condition, making their number independent of the coupling. These ideas where pioneered in [40]. Here, we will merely record the nature of the microscopic description of the D-brane systems in weak coupling for the cases considered without going into in any detail. Instead, the focus will be on the supergravity description of the system in strong coupling. This is done using the p-brane solutions as building blocks, following a few rules. The requirement of supersymmetry is satisfied when the p-branes intersect in an allowed manner, according to the rules in e.g. [42]. The general selection rule for a supersymmetric final solution is that the dimensionalities of the intersecting branes should differ by four [38], so that for example we get a 1/4 BPS state for a D1 brane lying inside a D5 brane, as each of the branes breaks half of the supersymmetry . The final requirement on the final solution is that of independence on the internal coordinates. In order to satisfy it, the position of each brane has to be ’smeared’, meaning that one should integrate over the possible position of every brane over the volume of the part of the internal manifold it is not wrapped on. This can be done in the higher dimensional setting by constructing a periodic lattice of branes along the directions that are to be compactified and then take the latice spacing to zero. Thus, for example, when wrapping a D1 brane over a cycle of a five dimensional compact manifold one has to consider modified harmonic function: Z 1 π 2 Q1 Q1 H1 = 1 + ⇒ H = 1 + . (7.1) 1 VM4 r6 2VM4 r2 The integral is over all of R4 and expresses the zero spacing limit of the lattice (this example will be used later). The general rule is that finally the Hp ’s for the branes become harmonic functions over the remaining noncompact transverse directions. This is explicit in the above example, since 1/r2 is a solution to the five dimensional Laplace equation. The final structure of the solution corresponding to a set of intersecting branes follows from the discussion of section 2 by iterating the procedure for single branes. This practically means that for each type of brane, the appropriate factors of these modified lower dimensional Hp ’s have to be placed in the metric components describing the directions along which the branes are wrapped or not, following (2.3). The same has to be done for the dilaton, where the various contributions multiply and for the p-form fields, where the different contributions add. Through the examples given below it will be seen that this procedure is very simple and intuitive. The charges of the black hole Another important point in the construction of black holes in string theory is the subject of the charges of the final object in the noncompact directions. Two cases of gauge fields are distinguished: the Kaluza-Klein gauge fields and the ones associated with higher dimensional tensor fields. According to a standard result, the charges for the Kaluza-Klein gauge fields are the momentum modes of the higher dimensional objects along compact directions generated by continuous isometries like circles, spheres etc - a familiar result. On the other hand, when some lower dimensional gauge fields are viewed as reductions of the ten dimensional p-form 39 fields, it follows that their charges will arise from the RR charges of the original p-branes. More concretely, the dimensional reduction of any string theory on a compact n-dimensional manifold Mn involves an expansion of the the p-form fields on the associated cohomology elements of that manifold. The relevant examples here are the ones that give 1-form gauge fields16 in the noncompact dimensions: X I aI . (7.2) F(p) ≡ dA(p−1) = F(2) A Here, aI are the elements of H p−2 (M ) (assuming that it is nontrivial) and I = 1, . . . , dim H p−2 . Then, the electric and magnetic charges (qI , pI ) of the lower dimensional theory are defined as (ommiting the normalization): Z Z I qI ∝ ?FI , p ∝ F I, (7.3) Σ Σ where F I are the field strengths defined above and Σ is a closed surface in the noncompact space enclosing the sources. Note that in general, the Hodge dual of F in the definition of the ∂L electric charge should be replaced by the dual field strength GI = ∂F I (L is the Lagrangian), hence the lower index. We suppress this feature for the sake of simplicity. Starting from this definition of the magnetic charges, Σ can be extended to a surface Σ̃ that incorporates the support of the aI ’s: Z I p ∝ F I aI (no summation) Σ̃ By the Poincaré duality between cycles and cohomology classes one can relate this to brane sources as follows. Given an m-dimensional cycle on Mn , one can define a m − n form involving its normal directions. In a convenient local coordinate patch with y m+1 . . . y n normal to the cycle this form will look like ρ(y)dy m+1 . . . dy n , where ρ(y) is a function integrating to one along the normal directions. One of the facets of Poincaré duality is that there is a one to one correspondence between m-cycles and n − m-degree cohomology classes. Thus, for any aI above, there is a cycle that lies in directions transverse to that form’s support and is contained in the interior of Σ̃. By using this, the sum of the magnetic equations in (7.3) P A is identical to the analogous ten dimensional equation for F(p) = A F(2) aA with magnetic sources along the Poincaré dual cycles of the basis forms, leading to the magnetic charges in (7.3). As the magnetic source of the p-form is an (8 − p) brane, it follows that this brane must be wrapped pI times on these cycles, giving rise to integer charges. By the same reasoning, the electric charges will arise from electric sources (p − 2 branes) wrapped on the cycles of the manifold complementary to the ones the magnetic sources are wrapped, in view of the presence the Hodge dual of F I in their definition. The final ingredient is the one to one correspondence of these electric and magnetic charges, owing to Poincaré duality that maps the cycles to cohomology classes H p−2 (M ) and H n−(p−2) (M ), 16 This is the only way to get vector fields in four dimensions, but not in higher dimensions as will be seen in the D1-D5 system 40 which are isomorphic for compact manifolds without boundary. Moreover, note that as the D-branes used are electric-magnetic duals to start with, the sum of their dimensionalities is constrained to be equal to six. By the construction above, this sum must be the dimension of Mn , which quite interestingly leads to a four dimensional noncompact theory in a string theory construction. In the M-theory case, one has M 5 and M 2 branes, so the compact manifold must be seven dimensional and the number of noncompact dimensions is the same. A final point that must be mentioned is the mechanism described in [43], which allows for a Dp brane to be viewed as an instanton of the worldvolume theory on a Dp + 4 brane. This is a second (much simpler) mechanism of creating charges for a black hole, which is behind the D1-D5 system in the next subsection. More generally, this enforces the presence of Dp − 4 brane charges when a Dp brane is considered, something that must be taken into account in constructing black holes from D-branes as will be seen later. The attractor mechanism Another issue is the fate of the scalars present in any supergravity theory in the final black hole solutions. One important aspect is that there should be no singularities, as that would signal a problem in the compactification. Supressing them is one of the reasons that more than one kind of branes are needed in the construction of black holes, as (some of) the moduli diverge at the horizon for a single brane. Thus, in a proper black hole solution, the values of the moduli are arbitrary in the asymptotically flat region and remain finite throughout the spacetime. But these finite values should not be random. Indeed, from a more physical point of view one can view the (arbitrary) values of the scalars at infinity as extra parameters characterising the black hole solution. If the near horizon geometry is allowed to depend on them in any continuous way, this will affect the entropy. But this is obviously a problematic case, as the entropy is the logarithm of the microscopic degeneracy of the system, an integer. Therefore, it should not vary continuously with the asymptotic values of the moduli. The solution to this problem is trivial for the hypermultiplet scalars, since the geometry (and thus the entropy) does not depend on the hypermultiplet fields. One can then consistently set the hypermultiplets to zero. For the vector multiplet scalars though, the problem persists, since supersymmetry relates them to the gauge fields, which are most certainly dynamical. Nevertheless, the physical expectation that the full solution near the horizon is described in terms of the charges only is still correct. This is accomplished through the attractor mechanism [62], which fixes the values of the scalars in the near horizon region in terms of the charges. According to this, the values of the moduli for a supersymmetric stationary solution are functions of only the radial distance from the horizon and their dynamics are governed by first order radial differential equations exhibiting attractor behaviour. Therefore, the net result of the flow from infinity to the horizon is to wash away almost all17 memory of the asymptotic values of the moduli as one approaches the horizon, making sure that the moduli eventually flow into constant values predetermined by the charges. In the following, we will repeatedly rely on this mechanism in order to avoid dealing with compactification 17 There is still some dependence left because not all of the moduli space is attracted into the same point. 41 scalars, in view of the fact that the AdS/CFT reasoning pursued here is based on the near horizon region. Although at first sight the methods and requirements for constructing black holes seem rather complicated, the following examples will show that in simple cases they are straightforward to apply. As will be seen, the practical application of the above boils down to a simple choice of D-branes with the right dimensionality to wrap the cycles involved, the rest being taken care of by the microscopic string theory constructions. 7.2 The D1-D5 system and five dimensional black holes The most studied example of a black hole solution constructed from D-branes is the D1D5-P system [40]. It is a simple D-brane model for a black hole in five dimensions based on D5 and D1 branes that does not make use of the general setting for the charges described above. Indeed, with only five compact dimensions the D5 branes wrap all of the compact space, so that there is no room left for the dual D1 branes. This is circumvented by using the fact that the two dual kinds of D-branes differ in dimensionality by four. As mentioned above, the D1 branes can be described as instantons in the worldvolume theory of the D5 brane [43], and this is what is used here to induce a D1 charge. More concretely, the system consists of N5 D5 branes wrapped on an internal five dimensional manifold M4 × S 1 (where M4 can be either T 4 or K3), and N1 D1 branes wrapped on the circle. Finally, Np units of left (or right) moving momentum are added on the resulting string wrapped along the circle. As the two kinds of branes break half of the supersymmetry through (2.1) and the momentum breaks all of the remaining left (or right) moving supersymmetries on the circle, this configuration preserves 1/8 of the ten dimensional supersymmetry. In the strong coupling limit, it describes a static black hole in the five noncompact dimensions, which can be viewed as a solution of 5D, N = 4 supergravity preserving 1/4 of the supersymmetry. The five dimensional charges arise as follows. First, there is one Kaluza Klein charge associated with the momentum on the circle. The D1 and D5 brane charges are the electric and magnetic charges of the RR 2-form field. The D5 charge descends to the electric charge of the gauge field in which the RR 2-form is dualised to after trivial reduction to five dimensions. On the other hand, the D1 charge is seen as the electric charge of the gauge field that results from reducing the RR two from along the circle18 . We briefly comment on the microscopic picture first. Assuming that the scale of M4 is much smaller than the radius of the circle, the zero modes of the system come from the vibrations of the effective string in six dimensions. Then, the low energy effective worldvolume theory is a 1 + 1 dimensional N = (4, 4) supersymmetric sigma model on the world volume of the string, but it is the superconformal theory to which it flows in the infrared that is used in actual calculations. This CFT is invariant in both the holomorphic and the antiholomorphic sectors under the so-called small N = 4 superconformal algebra, whose R-symmetry group 18 This somewhat confusing picture arises because there are two distinct ways of getting gauge fields in five dimensions, since in that dimension a two form field can also be seen as a gauge field 42 is SU (2) under which the four supercharges transform as two doublets19 . This R-symmetry will prove crucial for what follows. The end result for the entropy of the system for large charges is found by Cardy’s formula to be: p SD1D5 = 2π N1 N5 Np . (7.4) Note that the entropy depends on a multiplicative combination of the three charges, a manifestation of the fact that one needs at least three kinds of charges to get a nonvanishing macroscopic entropy in five dimensions20 . As an aside, note that a more robust description of the low energy theory can be formulated as a 1 + 1 dimensional sigma model with target space the moduli space of the instantons introduced above, namely the symmetric product (M4 )N1 N5 /S(N1 N5 ) [40], [27]. This is a more accurate description of the system and is the one used in actual computations. Turning to the supergravity description, according to (2.3) the solution of Type IIB supergravity associated to this system is (in string frame): r p 1 Qp H1 2 2 2 2 2 2 2 2 ds = √ −dt + dx + 2 (dx − dt) + H1 H5 dr + r dΩ3 + ds (M4 ), (7.5) r H5 H1 H5 H1 , (7.6) H5 where the Hodge dual is defined in six dimensions and εS 3 is the volume form on the unit sphere. The functions H1,5 are as in (2.6), but falling off as r−2 as is appropriate for a five dimensional transverse space: H(3) = dA(2) = 2Q5 εS 3 + 2Q1 e−2φ ? εS 3 , H1 = 1 + Q1 4G5 R N1 ≡ 1 + , πgs α0 r2 r2 H5 = 1 + gs α0 Q5 N5 ≡ 1 + , r2 r2 e2φ = HP = 1 + 4G5 NP QP ≡ 1 + , (7.7) πR r2 r2 implicitly defining the Q’s. Finally, R is the radius of the circle and G5 = 8π 6 gs2 (α0 )4 2πRVM4 (7.8) is the five dimensional Newton constant. Note the extra factors of volume in the Hp ’s with respect to (2.3) coming from G5 . These are a result of the smearing integrations, as explained above. The overall structure of the solution is of the form explained above with the various factors for the branes in the appropriate components of the metric. Note finally that the dilaton is regular everywhere, something that can not be arranged with only one kind of brane, as it would blow up21 at the horizon limit r → 0. In order to find the lower dimensional metrics associated with any compactification, a reduction along the compact directions has to be considered. The rule that take a string frame metric to an Einstein frame metric (the frame of interest here) are as follows: p 4 ds2E = e− d−2 φd ds2 , e−2φd = e−2φ detgint , (7.9) 19 There is also a ’large’ N = 4 algebra with R-symmetry group SU (2) × SU (2) × U (1) that is irrelevant in the present context 20 Momentum along a closed direction can be related through T- and S-duality to D-brane charge as well. 21 A D3 brane could do the job, as one can see from (2.3), but this possibility suffers from a zero area horizon problem 43 where φd is the d dimensional dilaton and gint is the internal metric. Using this to perform a reduction of the solution at hand over the directions spanned by M4 , one easily sees that in this case it corresponds to a mere omission of the M4 part in the metric. The above solution then becomes a black string solution of the six dimensional (2, 0) supergravity preserving half of the supercharges. This solution carries both electric and magnetic charges with respect to the six dimensional two form gauge field. It also carries momentum along the length of the string, leading to the extra charge in five dimensions. Note that (2, 0) supergravity arises in six dimensions in both the K3 and T 4 choices for M4 as only half of the supercharges of Type IIB supergravity are preserved by the reduction on either K3 or T 4 with a D5 wrapped on it, from the constraint (2.1). The five dimensional black hole metric is now easily found by a further dimensional reduction down to the five dimensional Einstein frame by the rules (7.9). The resulting metric is: ds2E = −(H1 H5 HP )−2/3 dt2 + (H1 H5 HP )1/3 dr2 + r2 dΩ23 , (7.10) with HP = 1 + QP /r2 . This is the metric of a static extremal black hole in five dimensions. In these coordinates it has a horizon at r = 0, whose area and Bekenstein-Hawking entropy are found to be equal to: A = 2π 2 p Q1 Q5 QP , ⇒ SBH = p A = 2π N1 N5 NP , 4G5 (7.11) when (7.7) is used. This exactly matches with the microscopic result (7.4). From this result it is obvious that the addition of the extra momentum charge on the string is crucial for the macroscopic result to be nonvanishing. In cases where fewer than three charges are nonzero the corresponding black hole has vanishing area and thus no macroscopic entropy, √ even though a microscopic entropy of the form Q1 Q2 is assigned to it. These are called ’small’ black holes and their interpretation is still under investigation. The solution just presented describes an extremal black hole in five dimensions. Although the tools to describe a general nonextremal black hole in string theory are not fully developed, we choose to use the expressions for the nonextremal case. This is done both for gain of generality and to avoid dealing with calculational subtleties of the extremal case that can obscure the line of thought. In fact, the plan is to consider Maldacena’s decoupling limit which will lead to (the much better understood) near extremal situations. Using the rules already described to find the nonextremal metric starting from the one above, one gets: h i p h i rH 1 µ 2 µ 2 1 2 2 2 2 − 1 − 2 dt̃ + dx̃ + H1 H5 1 − 2 dr + r dΩ3 + ds2 (M4 ), (7.12) ds = √ r r H5 H1 H5 H(3) = dA(2) = 2Q5 εS 3 + 2Q1 e−2φ ? εS 3 , e2φ = H1 , H5 (7.13) with Hi ’s and boosted time and x coordinate as in (2.13)-(2.16). This solution is not supersymmetric, as it is nonextremal. It can be described microscopically as a state of the (4, 4) CFT above that is excited in both the right and left moving sectors, whereas the extremal 44 solution was in the (supersymmetric) groundstate in the rightmoving sector. This will also be the interpretation in terms of the boundary CFT when the decoupling limit is taken. The charges of this solution are as in (2.15) for i = 1, 5, P and: X M =µ cosh 2ai , (7.14) i=1,5,P which in the extremal limit reduce to the three Qi ’s of the extremal solution above, with the P mass being the sum of them M = i Qi , as is appropriate for a BPS state. A further generalisation that will be briefly considered is that of a rotating black hole in five dimensions. The description in terms of D-branes is the same, but the D1-D5 system is now allowed to rotate in the four noncompact spatial directions. The Type IIB solution is the same as for a rotating black string in six dimensions (the metric of the four dimensional manifold is suppressed) [4]: 1 H1 H5 µfD r4 dr2 2 ds = √ −(1 − 2 )dt̃2 − dx̃2 + r fD (r2 + l12 )(r2 + l22 ) − µr2 H1 H5 − 2µfD 2µfD cosh a1 cosh a5 (l1 cos2 θdψ + l2 sin2 θdφ)dt̃ − 2 sinh a1 sinh a5 (l2 cos2 θdψ + l1 sin2 θdφ)dx̃ 2 r r 2 l2 µfD 2 2 2 2 2 2 + (1 + 2 )H1 H5 + (l1 − l2 ) cos θ( 2 ) sinh a1 sinh a5 cos2 θdψ 2 r r l12 µfD 2 2 2 2 2 2 + (1 + 2 )H1 H5 + (l2 − l1 ) sin θ( 2 ) sinh a1 sinh a5 sin2 θdψ 2 r r H1 H2 2 2 µfD 2 2 2 r dθ , + 2 (l2 cos θdψ + l1 sin θdφ) + (7.15) r fD with the Hp ’s and the tilded coordinates as in (2.13)-(2.16), and l1 cos2 θ l2 sin2 θ + . fD = 1 + r2 r2 (7.16) The li ’s parametrise the five dimensional angular momentum and are related to the spacetime SO(4) ' SU (2)L × SU (2)R charges JL,R as: ! Y Y µ JL,R = (l1 ∓ l2 ) cosh ai ± sinh ai (7.17) 2 i=1,5,P i=1,5,P This is again a nonextremal solution which breaks all supersymmetries. Note that the SU (2) charges will be related to the SU (2) R-symmetries of the right and left superconformal algebras in the microscopic theory. The rest of the charges are the same as for the nonrotating solutions. The extremal limit µ → 0, ai → ∞ of the above metric leads to a BPS state known as the BMPV black hole [41]. The right moving angular momentum JR vanishes, whereas the left moving one, JL , remains finite. This in line with the fact that one of the angular momenta of 45 a BPS state in five dimensions must be zero and the other can be arbitrary. The entropy of this black hole is found to be: r 1 (7.18) SBM P V = 2π N1 N5 NP − JL2 . 4 Note the rather unusual feature that adding angular momentum lowers the entropy of the system, which vanishes in the maximally rotating limit JL2 → 4N1 N5 NP . 7.3 Wrapped M 5 branes and four dimensional black holes Four dimensional black holes can also be modelled in Type II theories by wrapping branes on the compactification manifold, which we will take to be a Calabi-Yau (CY) manifold. In order to make the connection with M-theory easy, Type IIA string theory will be considered, so that only even dimensional branes can be wrapped on the CY. This gives two possible pairs according to the rules sketched: a D0-D6 and a D2-D4 pair. In fact, both must be used in order to get a black hole in four dimensions with a horizon area of macroscopic size. This can be seen heuristically from the fact that the D0 and D2 branes are worldvolume instantons of the D4 and D6 branes respectively, so that including one dual pair would generically induce (some combination of) charges of the other pair as well. For example, a system of D2-D4 branes will in general include D0 branes as worldvolume instantons on the D4 branes. In fact, this is a consistent reduction of the general D6-D2-D4-D0 system. A more general and useful picture is the M-theory lift of this system. Then, the D2D4 pair is a M 2-M 5 brane pair with the M 5 brane wrapped on the M-circle and the D0-D6 pair is now seen as electric and magnetic Kaluza Klein charges coming from the nondiagonal elements of the eleven dimensional metric along the M-circle. In fact, this is the most general configuration used in the microscopic description of all kinds of ’black object’ solutions for Calabi-Yau compactifications of M-theory, like black rings and supertubes, as will be seen later on. For a recent account see [50]. The entropy for the four dimensional black hole has not yet been accounted for microscopically in the general case, but only for zero magnetic Kaluza Klein (or D6) charge [9] and this will be the case considered here as well. The microscopic description [9] is based on the assumption of a M 5 brane wrapping the M-circle and the 4-cycles of the CY threefold. Then, M 2 brane charge is induced by turning on the three form field strength on the fivebrane worldvolume. Finally, momentum is added on the effective string along the M-circle. This configuration preserves 1/8 of the original supersymmetries as is appropriate to describe a solitonic object in the resulting N = 2 supergravity in the four noncompact dimensions. The charges arise in the standard way as wrapping numbers for M 5 and M 2 branes as already described. We therefore have as many electric charges qI as 2-cycles on the CY and as many magnetic charges pI as 4-cycles. As explained above, they are the same due to Poincaré duality and are equal to h1,1 , the (1, 1) Hodge number of the CY. To these, one must add the Kaluza-Klein electric charge q0 from the compactification on the circle (the magnetic one is set to zero). The low energy modes are governed by a (0, 4) supersymmetric 1 + 1 theory on the effective string worldvolume. 46 Again, this theory flows in the IR to a CFT whose rightmoving superconformal algebra is the small N = 4 algebra, as in the D1-D5 case, a property that will prove extremely useful. The microscopic entropy then arises as the logarithm of the number of states that preserve the right moving supersymmetry and have arbitrary excitations on the left moving side. This constrains the momentum along the circle in eleven dimensions to be leftmoving, or q0 > 0. The final result of the (rather involved) counting of the degeneracy of this system is the entropy: r q̃0 1 S = 2π (CIJK pI pJ pK + c2I pI ), q̃0 = q0 + DAB qA qB (7.19) 6 12 where DAB is the inverse of CIJK pK . In turn, CIJK are the intersection numbers of the Calabi-Yau manifold, defined through the basis elements aI of H 2 (CY ) by: Z CIJK = aI ∧ aJ ∧ aK . (7.20) CY Finally, c2I is the second Chern class of the tangent bundle on the Calabi Yau manifold, expanded on a basis of H 4 (CY ) that is Poincaré dual to the basis of two cycles used. On the macroscopic side, there are rules to find solutions of eleven dimensional supergravity corresponding to M-branes similar to the ones for Type II theories. In fact, the only changes are in the powers of the harmonic functions in the metric (as in (2.18)-(2.19)) and the nonexistence of the dilaton [42]. Unfortunately, the construction of the eleven dimensional metric is not as clear in this case due to the fact that the M 5 and M 2 brane charges do not represent stand alone branes, but arise as wrapping numbers and worldvolume excitations of a M 5 brane, so that only the result will be stated. The general structure of the solution is intuitively the one expected, with warp factors for each charge, but we refer to [45] for the details. Consider a M 5 brane wrapping pI times the 4-cycles of the CY manifold (labeled by I) and the M-circle and M 2 branes wrapping qI times the Poincaré dual 2-cycles. The numbers of these cycles are equal by Poincaré duality as explained above, so the same index is used. Furthermore, add q0 units of momentum along the resulting string wrapping the M-circle. The asymptotically flat solution of eleven dimensional supergravity corresponding to this situation is: (7.21) ds2 = F −2/3 −(dt + kdz)2 + F dz 2 + F 1/3 dr2 + r2 dΩ22 + ds26 , 1 F = C IJK ZI ZJ ZK , 6 k=− q0 X CIJK pI qI pI CIJK pI pJ pK + + 2 + , 3 r r 4r 24r JK (7.22) qI CIJK pJ pK + . (7.23) r 8r2 This solution exhibits a null stringlike singularity at r → 0. Note the extra term being added to the momentum charge q0 in the 1/r term of k. This will turn out to give the additive contribution in (7.19). The six dimensional metric has been omitted both because it is not known for a general CY and because it is not needed, since all the factors coming from the ZI = 1 + 47 presence of the branes conveniently cancel when dimensionally reducing. This can easily be seen in the trivial case CY = T 6 , where the six dimensional metric is: 1/3 1/3 1/3 Z2 Z3 Z1 Z2 Z1 Z3 2 2 2 2 2 ds6 = dx2 + dx3 + dx1 + dx2 + dx21 + dx23 . (7.24) 2 2 2 Z1 Z3 Z2 By reducing over the M-circle by the rule given below in (7.31), and making use of the string theory dimensional reduction rules in (7.9), one can verify that all the Zi ’s cancel. As mentioned above, there is a nontrivial expression for the three form field. This can be given in terms of the gauge fields obtained by expanding it on H2 (CY ), which is usefull when discussing the solution from a five dimensional point of view. Unfortunately though, the related expressions are given implicitly though differential equations and we refrain from showing them. What is going to be of importance to the following developments is that in the decoupling limit the electric components of the solution in [45] are subleading to the magnetic ones as one approaches the boundary of the resulting AdS3 × S 2 geometry. In that case, the field strengths take the form: 1 F I dec = − pI S 2 , (7.25) 2 where S 2 is the volume form on the unit two sphere. This shows that the asymptotic form of the solution at the boundary is that of a Freud-Rubin geometry with two form flux on the sphere. For later reference, we briefly discuss the expressions for the moduli in the five dimensional theory obtained after compactifying on the CY. This results to a black string solution of five dimensional N = 1 supergravity for which the moduli are constrained by the attractor mechanism to take fixed values at the near horizon region. As the focus will be on this region, we restrict attention to these constant values: XI = pI . ( 16 CIJK pI pJ pK )1/3 (7.26) The index I runs from 1 . . . dim H 2 (CY ), i.e. there is one of them for each gauge field coming from the expansion of the three form of the eleven dimensional theory. These real scalars represent the volumes of the various two-cycles of the CY and will not be of much importance in the following. The supersymmetries preserved by this solution can be found by considering the black string solution of the five dimensional theory and examining its explicit supersymmetry transformation rules. The result is that it preserves (0, 4) supersymmetry on the black string worldvolume, as expected from the microscopic construction. This can be seen in a less cumbersome way though the eleven dimensional solution in the limiting case of the Calabi-Yau manifold being just the six torus as in [45], where the solution (7.21) was constructed. Then, the solution will correspond to three M 5 branes wrapping the four cycles of T 6 and three M 2 branes wrapping its T 2 ’s. This is a singular limit of the original case where the M 5 brane wrapped the four cycles of a general Calabi-Yau manifold. If the torus is chosen to span the (123456) directions, the eleven dimensional projections imposed by the presense of the three 48 M 2 branes wrapping the 2-cycles on the six torus are: (1 + Γ012 )η = (1 + Γ034 )η = (1 + Γ056 )η = 0, (7.27) just like the D-brane case. It is a straightforward exercise in gamma matrices manipulation to show that these equations imply that Γ1234 η = Γ1256 η = Γ3456 η = η, (7.28) by using the fact that Γ11 = Γ0 . . . Γ10 = 1 in eleven dimensions, a relation that will prove important again later. Additionally, the presence of the M 5 branes imposes similar relations: (1 − Γ0z1234 )η = (1 − Γ0z3456 )η = (1 − Γ0z1256 )η = 0, (7.29) which along with the property in (7.28) become: (1 − Γ0 Γz )η = 0. (7.30) We thus see that the four supersymmetries preserved by the black string solution are right moving with respect to the z direction, the result stated above. The final four dimensional black hole metric is found by dimensionally reducing to four dimensions. A first reduction to ten dimensions can be done by the rule: 2 ds211 = e−2φ/3 ds210 + e4φ/3 dz + A0µ dxµ , (7.31) to get the ten dimensional dilaton and the Kaluza-Klein gauge field associated with the D0 and D6 brane charges: 2 k (F − k 2 ) 3 2φ , A0 = − dt. (7.32) e = F F − k2 The further reduction down to four dimensions is done by the rules shown in (7.9). The resulting metric in Einstein frame reads: ds2E = − √ √ dt2 + F − k 2 dr2 + r2 dΩ22 , F − k2 (7.33) which is easily seen to describe an extremal black hole. The horizon is located at r = 0 and its area results to the entropy in (7.19) (in units where G4 = 1/4), but without the subleading correction proportional to c2 . A match with the full microscopic result requires the inclusion of higher derivative corrections in the supergravity action. This was worked out in a series of papers [11] from a purely four dimensional supergravity point of view and will be reproduced in the following through the use of the near horizon AdS3 region. 7.4 Black rings in five dimensions As the subject of black holes started in the physical case of four dimensions, an extensive research of this case has led to the famous uniqueness theorems for black holes. Under some 49 rather general physical assumptions, these theorems show that the only relevant topology of a black hole horizon in four dimensions is that of R × S 2 . This however is not the case in higher dimensions, where more ’exotic’ objects can be constructed. In five dimensions, one encounters the black ring solution [56], with horizon topology that of R × S 1 × S 2 . Thus, when dealing with more than four dimensions, one does not have a unique black hole corresponding to a set of charges, but a finite number of black objects with different horizon topologies. From a microscopic standpoint one can think of these situations as different ’phases’ of the same system. Here, we will consider a version of the black ring embedded in five dimensional supergravity [57]. This solution can be seen from an M-theory point of view as the result of a set of M 2 branes wrapping 2-cycles and M 5 branes wrapping 4-cycles of a Calabi-Yau manifold which makes up the extra six dimensions. The remaining string like object in five dimensions (the part of the M 5 branes not wrapped) carries angular momentum and is thought of as forming a closed loop as in the previous section, giving rise to a ring-like horizon. This description is exactly the same as for the four dimensional black hole, so it follows that it preserves the same number of supersymmetries, namely four, and that the microscopic description will be the same as in the previous section. In fact, the only difference with the black string solution above is the addition of angular momentum in the second subgroup of the SO(4) ∼ = SU (2)L × SU (2)R rotation group. This extra momentum supports a horizon of the form S 1 × S 2 with nonzero area. However, an important subtlety should be taken into account, namely that in the present case the M 5 branes are contractile in the five noncompact dimensions. This simply means that there are really no conserved magnetic charges associated with them from a purely five dimensional point of view. In fact, a macroscopic observer in five dimensions would see a closed electrically charged loop with a current flowing around it, giving rise to a magnetic dipole moment. We now present the solution, assuming that the six compact dimensions have the geometry of T 6 . The metric takes the form: ds211 = ds25 + X 1 (dx21 + dx22 ) + X 2 (dx23 + dx24 ) + X 1 (dx25 + dx26 ), (dt + ω)2 R2 (H1 H2 H3 )1/3 dy 2 dx2 2 2 2 2 2 ds5 = − + + (y − 1)dψ + + (1 − x )dφ , (7.34) (x − y)2 y2 − 1 1 − x2 (H1 H2 H3 )2/3 where the functions X i and Hi are given by: qi − i|jk| pj pk i|jk| pj pk 2 Hi = 1 + (x − y) − (x − y 2 ), 2R2 4R2 (H1 H2 H3 )1/3 X = Hi i (7.35) and the one form ω = ωφ dφ + ωψ dψ by: ωφ = − 1 (1−x2 ) pi qi − p1 p2 p3 (3 + x + y) , 2 8R 1 1 y2 − 1 ωψ = (p1 +p2 +p3 )(1+y)− 2 ωφ (7.36) 2 8R 1 − x2 The notation |ij| means sum with i > j. The coordinates used are the so-called ring coordinates, which are designed to foliate flat four dimensional space with ringlike surfaces. 50 These surfaces are labelled by the coordinate y, which takes values −∞ < y < −1, with asymptotic infinity lying at y = −1 and the innermost surface y = −∞ being a circle. The other three coordinates are angular coordinates parametrising the rings of topology S 1 × S 2 , with 0 < ψ < 2π parametrising the circle and 0 < cos−1 x < π, 0 < φ < 2π parametrising the sphere in the usual way. The four dimensional metric in square brackets in (7.34) along with the factor of (x − y)2 in front is in fact the flat four dimensional Euclidean metric in these coordinates. The present case of the CY being taken as T 6 is referred to as the U (1)3 model, as the eleven dimensional three form gives rise to three U (1) gauge fields labelled as Aiµ , corresponding to its expansion with respect to the three harmonic (1, 1) forms on T 6 . These fields take the following form in this solution: Ai = Hi−1 (dt + ω) − pi [(1 + y)dψ + (1 + x)dφ] . 2 (7.37) Note that both here and in the metric the familiar factors of Hi associated with the different M 2 branes are found, differing only in the coordinates used. As the reader might have read off from the notation by now, qi and pi are the wrapping numbers of the electric M 2 and magnetic M 5 branes, giving rise to the electric charges and magnetic dipole moments for the three gauge fields in five dimensions22 . The mass M , angular momenta Jφ , Jψ , electric charges Qi and magnetic dipole moments Di are given by the following formulae: M= π (q1 + q2 + q3 ), 4G5 π (qi pi − p1 p2 p3 ), 8G5 Jφ = 1 Qi = 16πG5 ?F i π = qi , i 4G5 S3 X Z π (2R2 (q1 + q2 + q3 ) + qi pi − p1 p2 p3 ) 8G5 Z 1 pi i D = Fi = . (7.38) 16πG5 S 2 8G5 Jψ = We see that the solution is parametrised by seven quantities, namely the electric charges, the magnetic dipole moments and the ring radius R. Note that a nonzero ring radius is controlled by the difference Jψ − Jφ . It will prove convenient to trade R with Jψ in the following. Note finally that the mass is just the sum of the three conserved charges, as it should be for a BPS state. We end this very brief account of the properties of the supersymmetric black ring by giving the entropy associated with it. Both the macroscopic and microscopic [58] pictures give the result: s 1 2 3 ppp 1 IJ 1 2 3 S = 2π p p p − Jψ + D qI qJ , (7.39) 4 12 which is exactly (7.19) when the Calabi-Yau manifold is the six torus as expected, since the 3 microscopic is identical. In this case, c2 = 0 by definition, CIJK = 6δI1 δJ2 δK and the matrix IJ −1 K −1 D = (DIJ ) ≡ (CIJK p ) . 22 Note that the T 6 is decomposed into three T 2 ’s, accompanied by their complex structures, so that there are indeed only three independent (1, 1) forms. Their Poincaré duals define a corresponding basis of (2, 2) harmonic forms. 51 8 AdS/CFT for black holes In the present section, the proposal for the determination of the entropy by use of the AdS/CFT correspondence is presented. It is based on the decoupling limit of the black hole solutions for the examples in section 7, which in all cases leads to a locally AdS3 factor. This is in line with the AdS/CFT correspondence, as in all the examples given the worldvolume theory on the D-branes can be accurately approximated as a 1 + 1 CFT. The entropy is then found through the relation in section 6 provided that the central charge is known. In fact, finding the central charge is the main goal in the examples below and is achieved by relating it through the supersymmetry algebra to the coefficient of the Chern-Simons terms in the bulk along the lines of section 4. We proceed in turn with the treatment of the D1-D5 system and the wrapped M 5 brane in this framework, following the exposition in [5], [6]. A few comments on the supersymmetric black ring in five dimensions are included in the end. 8.1 D1-D5 system We present an analysis of the D1-D5 system from the point of view of its decoupling limit and the CFT induced on the near horizon AdS3 region. We begin with a discussion of the decoupling limit of the solution in (7.12) and the resulting AdS3 factor of the near horizon geometry. Upon use of the simple Brown-Henneaux result, the entropy of both the static and rotating five dimensional black holes is retrieved. Then, a dimensional reduction of the relevant part of the six dimensional (2, 0) supergravity action on AdS3 × S 3 is performed and the three dimensional Chern-Simons terms are identified. These terms are then used to find the central charges of the boundary CFT, which in this case coincide with the BrownHenneaux central charge associated to the asymptotically AdS3 factor. Finally, an important relation of the result with the anomalies of the microscopic CFT is discussed. 8.1.1 Physics in the decoupling limit The decoupling limit for the D1-D5 system is easily found from the solution (7.12) by dropping the constant from the harmonic functions associated with the brane charges, as explained in detail in Appendix A. Note that this is necessarily a near extremal limit, as discussed there, so that any result that follows is only applicable to that case. The result is of the form: s h i p dr2 p 2 r µ µ Q1 2 2 − 1 − 2 dt̃2 + dx̃2 + Q1 Q5 1 − 2 + Q Q dΩ + ds (M4 ), ds2 = √ 1 5 3 r r r2 Q5 Q1 Q5 H(3) = dA(2) = 2Q5 εS 3 + 2Q5 ? εS 3 , e2φ = Q1 . Q5 (8.1) Recall that t̃, x̃ are related to time and the coordinate along the string through (2.16). A few comments are in order. First, it is evident that the space decomposes as AdS3 × 3 S × M4 , with the AdS3 spanning the t, x, r directions. In fact, under the assumption of large 52 charges, the relation between the length scales of the three spaces is lAdS = lS 3 lM4 , where lAdS , lS 3 , lM4 are the radius of AdS3 , the radius of the S 3 and the length scale of M4 respectively. This shows that the near horizon region is accurately described as a six dimensional geometry containing a black string, which factorizes as an asymptotically AdS3 space times the three sphere. Therefore, it can be studied in AdS3 supergravity with Kaluza-Klein fields coming from the S 3 fiber. The AdS3 × S 3 near horizon region is a higher dimensional generalization of the near horizon region of the extremal black hole, which is of the form AdS2 × S 3 in five dimensions. What has happened here is that the near horizon geometry has ’combined’ with the compactification circle to give an AdS3 factor, even for the nonextremal case. Note also that in this nonrotating example the decomposition is global, but when rotation is added to the black hole, one does not get just the sphere metric. Instead, the usual Kaluza-Klein ansatz gij (dxi − Aiµ dxµ )(dxj − Ajµ dxµ ) arises, and the corresponding gauge fields are excited in the AdS3 base. We will consider this case briefly in the following. Let us apply the content of the AdS/CFT correspondence to the D1-D5 example shown here. The near horizon AdS3 × S 3 supergravity is supposed to be dual to the 1 + 1 dimensional CFT of the microscopic description. In both cases we are neglecting the M4 manifold assuming that it is small compared to the other length scales. Note the matching of the symmetries in the correspondence, starting from the AdS3 isometry group SO(2, 2), which maps to the (globally defined part of the) conformal group in two dimensions. In both cases there is an SO(4) symmetry, which in the near horizon region comes from the S 3 fibration and in the CFT on the D1-D5 intersection from the rotations in the four spatial transverse flat directions and is identified with the SU (2) × SU (2) R-symmetry of the (4, 4) worldsheet theory [27]. Finally, the number of supersymmetries is again the same, as the 16 supersymmetries of the CFT match with the 16 supersymmetries preserved by the maximally symmetric near horizon region (see next subsection for a justification of this). In fact, this 1 + 1 CFT can be thought as living at the boundary of the asymptotically AdS3 space as explained, so that its properties can be inferred using the methods already set up in the previous sections. In particular, the central charge and the eigenvalues of the right and left moving Virasoro operators are relatively easy to compute and can lead to a calculation of the entropy of the black hole. We start with the Virasoro operators L0 , L̃0 , which are related to the mass and angular momentum of the locally AdS3 space through (4.31). The values of these charges can be made manifest by rewriting the AdS3 part of the near horizon metric (8.1) as the BTZ black hole using the change of radial coordinate ρ2 = r2 + µ sinh2 aK and an appropriate rescaling of all coordinates by R (R is the radius of the circle along the string): 2 ρ4 − 8G3 l2 M ρ2 + 16G23 l2 J 2 2 l2 ρ2 dρ2 4G3 J 2 ds = − dt + 4 + ρ dφ − dt . l 2 ρ2 ρ − 8G3 l2 M ρ2 + 16G23 l2 J 2 ρ2 (8.2) √ 2 Here, l = Q1 Q5 , G3 is the three dimensional Newton constant and: 2 M= R2 µ cosh 2aP , 8G3 l4 J= R2 R2 µ sinh a cosh a ≡ QP . P P 4G3 l3 4G3 l3 53 (8.3) The value of the three dimensional Newton constant can be related to the five dimensional one by comparing compactifications from six dimensions: G6 = G3 V (S 3 ) = G5 2πR ⇒ G3 = G5 2πR . 2π 2 (Q1 Q5 )3/4 (8.4) With this information and assuming that the central charge of the boundary CFT is given by 3l the naive Brown Henneaux relation c = 2G we can immediately compute the entropy from 3 Cardy’s formula: r r c c c c S = 2π (L0 − ) + 2π (L̃0 − ) = (8.5) 6 24 6 24 r r π2 π2 Q1 Q5 Q1 Q5 µ (cosh 2aP + sinh 2aP ) + µ (cosh 2aP − sinh 2aP ). (8.6) 2G5 4 2G5 4 The extremal limit is µ → 0, aP → ∞ with (8.7) QP = µ sinh aP cosh aP , held fixed, as required by (2.15). The entropy becomes: S= π2 p Q1 Q5 QK , 2G5 (8.8) which agrees with (7.11). We therefore see that the CFT at the AdS3 boundary gives the correct entropy and can also predict a value for the nonextremal case. It should be noted though that this result is valid only for near extremal black holes, as the decoupling limit necessarily leads to the near extremal region (see the discussion in Appendix A). This result can be easily extended to the rotating case, by taking into account the extra charges with respect to the SO(4) ∼ = SU (2) × SU (2) Kaluza Klein gauge fields and using the expression (6.14) for the entropy of a charged black hole. One has to take the decoupling limit of the metric in (7.15) using the rules in Appendix A: r2 l1 l2 (r2 + l12 )(r2 + l22 ) − 2µr2 2 √ √ dt̃ + (dỹ − 2 dt̃)2 + ds = − 2 r Q1 Q5 r Q1 Q5 √ h i p Q1 Q5 r2 2 2 2 2 2 2 dr + Q Q dθ + sin θd φ̃ + cos θd ψ̃ , 1 5 (r2 + l12 )(r2 + l22 ) − 2µr2 2 with dφ̃ = dφ − √ 1 (l2 dt̃ + l1 dỹ), Q1 Q5 dψ̃ = dψ − √ 1 (l1 dt̃ + l2 dỹ). Q1 Q5 (8.9) (8.10) The only new feature here is the presence of the Kaluza-Klein gauge fields coming from the nondiagonal components of this metric. We thus have again a locally AdS3 × S 3 space but globally the fibration is nontrivial, unlike the direct product nature of the decoupling limit of the nonrotating metric. In the large D-brane charge limit and by use of (7.17) and (3.10) the KK gauge fields and their associated charges are found to be: 1 AL,R = √ (l1 ∓ l2 )(cosh aP ± sinh aP )d(x ∓ t) 2 Q1 Q5 54 ⇒ KK JL,R = kAL,R = JL,R , (8.11) as expected. We refer to Appendix B for the definition of the right and left moving gauge fields contained in SO(4) ∼ = SU (2)L × SU (2)R . Now, (6.14) can be directly used with the explicit charges and the gauge part of L0 omitted, since this is a situation with constant gauge fields as described in connection with that equation. As the scale of the AdS3 space is the same as before and using the simple Brown-Henneaux central charge again, only the asymptotic mass and angular momentum of the AdS3 part of the metric are needed. After substituting (2.16) in (8.9) and rewriting it as a BTZ metric, these charges are read off as: R2 M= (2µ − l12 − l22 ) cosh 2ap + 2l1 l2 sinh 2ap , Q1 Q5 J= R2 2 2 (2µ − l − l ) sinh 2a + 2l l cosh 2a , p 1 2 p 1 2 8G3 (Q1 Q5 )3/4 (8.12) where R is the radius of the compactification circle from six to five dimensions. By use of (8.4) and (6.14) and with the convention G5 = π/4, the entropy takes the form: hp i p p (8.13) S = π N1 N5 2µ − (l1 + l2 )2 e−ap + 2µ − (l1 − l2 )2 eap , which matches with the macroscopic result for large charges [4]. Upon taking the extremal limit, the entropy of the BMPV black hole [41] emerges: r 1 Srot = 2π N1 N5 NP − JL2 , (8.14) 4 as expected. 8.1.2 Compactification on S 3 Having stated the result, we now turn to the analysis of the properties of the CFT at the AdS3 boundary, in order to show that there are no extra contributions from gauge or gravitational Chern-Simons terms which would change the simple Brown-Henneaux central charge used above. The results reviewed here are the following. The boundary CFT is a (4, 4) sigma model, so that supersymmetry can be used to relate the central charge to the level of the SU (2) algebras. To be more specific, each of the right and left N = 4 superconformal algebras includes an SU (2) Kac-Moody algebra, whose generators rotate the supercharges into each other as two doublets. The level k of this algebra is constrained to be proportional to the central charge c of the Virasoro part: c = 6k. Now, the operators in the SU (2) algebras of the CFT are related to the SU (2) Kaluza-Klein gauge fields in the AdS3 bulk upon reduction on AdS3 × S 3 through the AdS/CFT correspondence. It follows that the levels of the boundary SU (2) algebras must be equal to the coefficients of the associated three dimensional Chern-Simons terms because of (4.16), uniquely determining the central charge. In fact, after calculating the Chern-Simons terms for one of the right or left moving parts to find one of the central charges, the other can be found if the coefficient of a possible 55 gravitational Chern-Simons term is known, as in (4.34). In the present case there is no leftright asymmetry, so that the right and left central charges in the microscopic CFT are equal. Therefore, no gravitational Chern-Simons terms should appear in the three dimensional action. This will prove to be the case in what follows for the tree level Type IIB supergravity compactified on AdS3 × S 3 × M4 . Of course, this has to be confirmed by adding higher derivative corrections, since they are the ones that are generally responsible for gravitational Chern-Simons terms. We now proceed in justifying the above statements. First, we take the time to show that the AdS3 × S 3 compactification considered is a maximally supersymmetric background of six dimensional (2, 0) supergravity. This follows from the fact that in the background in (8.1), any spinor on the sphere can be used in a decomposition of the six dimensional supersymmetry variation parameters. Writing the six dimensional spinor as ξ = (xµ ) ⊗ η(y m ), where , η are spinors on the AdS3 and S 3 respectively, the sphere part of the six dimensional variation of (any of the) the gravitino(s) of six dimensional supergravity becomes: 1 δψM = DM (ω)ξ − GM N P ΓN P ξ = 0 4 ⇒ 1 δψm = Dm (ω)η − Gmnp Γnp η = 0 4 (8.15) Here, D(α) is the covariant derivative with connection α and ω is the ordinary spin connection. Note that in the first (second) equation the indices on the gamma matrices are the six (three) dimensional ones. The last equation can be written more concisely as Dm (ω − 41 G · Γ)η = 0. The integrability condition for it then becomes: 1 1 1 ab ab 1 [Dm (ω − G · Γ), Dn (ω − G · Γ)]η = 0 ⇒ Rmn Γ η + Gmqr Gnps [Γqr , Γps ]η = 0 ⇒ 4 4 4 16 1 1 ab Rmn Γab − Gmqr Gnps qrk psl Γkl η = 0, (8.16) 4 4 where we used that Γab = iabc Γc in three dimensions. For the background (8.1) this vanishes identically if one remembers to rescale Gmnp by a factor of eφ according to (2.2) (note that the dilaton is constant in the near horizon region due to the attractor mechanism). This means that any spinor on the three sphere is good enough, so that all the supercharges of the original six dimensional theory survive in three dimensions. In particular, a spinor η in six dimensions can be chosen to be chiral with four complex components. Furthermore, its eight degrees of freedom can be arranged into a pair of real four component spinors, of which none constitutes an irreducible representation of the six dimensional Lorentz group. Their usefullness is that they span the positive and negative definite subspaces of a symplectic space, motivating their characterisation as symplectic Majorana spinors. Decomposing them in terms of 2 ⊗ 2 spinor in the two spaces will then give four real spinors in three dimensions. In order to find how they transform under the SO(4) rotations on the sphere, consider the decomposition of the full SO(5, 1) spinor above under SO(1, 1) × SO(4) ∼ = U (1) × SU (2)L × SU (2)R (the symmetries on the string and the transverse rotations): 1 1 4 −→ [ , (2, 1)] ⊕ [− , (2, 1)]. 2 2 56 We see that the left (right) moving supercharges transform under the left (right) SU (2) rotations, in accordance with what was said above for the R-symmetry of the microscopic theory. When the 4 spinor is further broken down to symplectic Majorana spinors, the two representations on the right will give the four spinors in three dimensions. Due to their origin, they will transform as two doublets under the SU (2)L and SU (2)R groups respectively. Now, the original six dimensional supergravity has two such chiral supersymmetry parameters which can be arranged in four symplectic Majorana spinors, with R-symmetry group U Sp(4) = Spin(5) that rotates them. Under the Kaluza-Klein decomposition, they will result in twice the number of spinors found for one of them above. Then, one finds two SU (2) doublets of spinors in three dimensions which span the positive subspace of the U Sp(4) group and two more spanning the negative subspace. Thus, the expected three dimensional SU (2)L × SU (2)R symmetry arises by the inequivalence of the rotations of the two pairs of six dimensional spinors, as spinors in a different subspace of the symplectic vector space must transform under a different SU (2) R-symmetry. The main reason for this digression is that now it is clear that the three dimensional theory should be invariant under a pair of identical, decoupled right and left supergroups containing eight supercharges, whose bosonic subgroup must be SO(2, 1)×SU (2). The unique such supergroup is SU (1, 1|2). More on this decomposition of the supersymmetries can be found in [35], where the three dimensional theory is identified with the Chern-Simons SU (1, 1|2)L × SU (1, 1|2)R supergravity. We now turn to the explicit construction of the ChernSimons terms associated with the Kaluza-Klein gauge fields. In the specific background (8.1) considered, the dilaton is constant, so that the only nontrivial fields are the metric and the RR two form gauge field, whose flux supports the sphere. This type of solutions are referred to as Freud-Rubin [52] type solutions. In order to exhibit the Chern-Simons terms in the AdS3 reduction, we consider the simplified setting of a theory with only these two kinds of fields: Z 1 1 6 S=− d x R ?1 + ? H ∧ H , (8.17) 16πG6 2 where GM N P is a three form field strength, ? denotes the Hodge dual in six dimensions and ?1 is shorthand for the six dimensional volume form. Note that in order for this to describe the solution in (8.1), a constant equal to e2φ coming from the compactification of Type IIB supergravity on M4 should stand in front of the gauge kinetic term, but is omitted for brevity. It will be reinstated at the end of the calculation. For a more general discussion with the same result, see [59], where a dynamical dilaton is kept. As in the particular solution (8.9), the reduction of this action on S 3 will have Kaluza-Klein excitations, as the space decomposes only locally as an AdS3 space times the three sphere. Since our task is to find the Chern-Simons terms for these gauge fields in the AdS3 reduction, we will use the local ansatz: ds26 = ds23 (AdS3 ) + l2 δij (dxi − Aik xk )(dxj − Ajl xl ), (8.18) with Am depending only on the AdS3 base coordinates, i, j, . . . = 1, . . . , 4 and the constraint P i i x x = 1 is assumed. In the following, we will assume that the metric takes this form 57 asymptotically near the AdS boundary, with no reference to the inner bulk. The gauge transformations relevant to the reduction in (8.18) are: δAij = DΛij . δxi = Λij xj , (8.19) The covariant derivative is D = d − A and is used to define the vertical forms Dxi = dxi − Aij xj and field strengths: Fji = [D, D]ij = dAij − Aik ∧ Akj , (8.20) as covariant objects. The only part missing now in order to generalise (8.1) is an ansatz for the three form field strength. There are several choices known in the literature. In any case, by the requirements of closure and equality to the form in (8.1) when Aij = 0, it must contain the volume forms on the sphere and AdS3 parts plus some closed forms. These are chosen on the basis of extra requirements, the most important of which is that its integral on the sphere should be equal to Q5 as in (8.1), expressing the total charge of the configuration. Now, one can impose that the ansatz be also gauge invariant, a generally desirable feature. A form satisfying this can easily be written down using the vertical forms Dxi above: 1 1 i j k 1 ij k 2 l H = 2Q5 2π e3 + ?S 3 , e3 = ijkl x Dx Dx Dx − F Dx , (8.21) (2π)2 3 2 with S 3 the volume form on the unit S 3 . A related form has appeared in [35]. Although gauge invariance is an aesthetically nice feature, it clashes in this case with another desirable property of the ansatz, namely the property of a well defined gauge field. This is by itself not absolutely crucial, but since the D-branes in the problem couple directly to it, it should be well defined. Moreover, it is usefull for anomaly cancellation as will be seen in the next subsection. But as can be seen from (8.21), a possible two form B for which dB = H, would have to be expressed through nonlocal terms since terms with no derivatives appear. In order to get a well defined two form gauge field, the requirement of gauge invariance has to be dropped. This may seem to be a pathological situation in the context of dimensional reduction, but in fact, as shown in [13], it only leads to an action that varies by a boundary term under gauge transformations, so that it can actually give well defined equations of motion. The ansatz takes the form: (8.22) H = 2Q5 2π 2 (e3 − χ3 ) + ?S 3 , where χ3 is a closed three form, proportional to the difference of two Chern-Simons forms for the Kaluza-Klein gauge fields for the right and left SU (2) rotations in SO(4): χ3 = − 2 1 2 1 T r(AL dAL + A3L ) + 2 T r(AR dAR + A3R ) 2 8π 3 8π 3 (8.23) Explanations on this and other formulae related to the reduction on S 3 are collected in an Appendix. In (8.22) all noninvertible terms cancel, and the explicit form of the two form gauge field can be written down [13]. Moreover, as mentioned above, this same ansatz has been derived in [59] and shown to lead to a consistent reduction on S 3 . This provides a stronger basis for this unfamiliar choice. 58 As can be seen from the explicit form of e3 , it involves the volume form on the sphere, a few cross terms with legs in both AdS3 and the sphere, and two AdS3 three forms of the schematic structure AF and A3 , like the terms contained in the Chern-Simons form. When (8.22) is used in the action, where the form ?H ∧ H is integrated over the full six dimensional space, the presence of Chern-Simons forms in three dimensions is evident from the cross terms between the sphere volume form in (8.22) and the term e3 − χ3 : 8Q25 S 3 ∧ 2π 2 (e3 − χ3 ) . 32πG6 (8.24) As this is proportional to the volume form on the sphere, the only terms to survive the sphere integration will be the full χ3 term and the Chern-Simons-like terms in e3 pointed out above. It is easy to verify the absence of any more terms of this form that could cancel these ones, the other parts of ?H ∧ H giving eventually other three dimensional terms. Note that the χ3 contribution is already written as a sum of right and left Chern-Simons terms, so we will concentrate on the contribution from the e3 term and show that it takes the same form. Since we will not be interested in Yang-Mills terms at all, and for ease of computation, we will make use of the gauge condition employed in sections 3 and 4, namely setting the radial component of Aµ to zero, at least asymptotically near the boundary. Note that this situation is the one generally expected when a higher dimensional rotating solution is reduced along the rotating directions, as in (8.9). This choice cancels all terms in e3 − χ3 of order higher than two in A, which in general will contribute to both Yang-Mills and Chern-Simons terms in three dimensions. Relying on the fact that the final result must be expressible through these two terms (the only ones allowed in three dimensions), we will go after the terms of the form AdA only. With this out of the way, consider the AF term in e3 with its precise coefficient in the expression (8.24) (with no A3 terms): Z 2Q25 − S 3 ∧ ijkl xl dAij Akm xm . (8.25) 32πG6 AdS3 ×S 3 Because of homogeneity of the sphere, the term xm xl can be replaced by 41 δ ml under the sphere integral. Then, the overall integral over the sphere gives a factor of 2π 2 , the final result being: Z Z Q25 π 2 Q25 π 2 ij km − ijkl dA A = T r (AL dAL − AR dAR ) (8.26) 64πG6 AdS3 16πG6 AdS3 We refer again to the related Appendix for an explanation of the transition to the form with the right and left SU (2) gauge fields. The contribution from χ3 is already in the most convenient form and only its numerical coefficient is of interest. As can be easily seen by replacing the form of χ3 in (8.24), it is equal to the contribution from e3 just found. Adding these two contributions to the Chern-Simons terms, the final result takes the form: Z π 2 Q25 T r (AL dAL − AR dAR ) . (8.27) 8πG6 AdS3 59 This expression (when the A3 terms are included) is a sum of two Chern-Simons terms as the ones studied in sections 3 and 4, with Q25 2G6 2φ π k=e 2 ⇒ k = N1 N5 , (8.28) after reinstating the dilaton factor. In the second equality we used the equations in (7.7) and the relation between the five and six dimensional Newton constants G6 = 2πRG5 , with R the radius of the compact sixth dimension. This will lead to the same central charge c = 6N1 N5 used before, as will be seen in a moment. First, note that there is no unexpected right/left asymmetry coming from potential gravitational Chern-Simons terms in the three dimensional action. This is impossible because the gravitational part of the six dimensional theory is the standard Einstein-Hilbert term which is known not to produce such terms when dimensionally reducing. It then follows that the two central charges of boundary CFT will be equal. The Chern-Simons we did find in (8.27), induce a leftmoving and a rightmoving affine SU (2) algebras with level k = N1 N5 , along the lines of the discussion in section 4. To go a step further, the full algebra induced on the boundary can be found by clarifying the field content of the AdS3 theory and using supersymmetry. In the present case, a consistent action for the massless fields in three dimensions can be written down, if compactification moduli are included [59]. Moreover, by the result of [23], any AdS supergravity in three dimensions can be written as a Chern-Simons supergravity for the right and left supergroups. Taking into account the symmetries explained above, it follows that the full action for the massless fields is the SU (1, 1|2)L × SU (1, 1|2)R Chern-Simons supergravity coupled to the compactification moduli. But as the scalar fields in the near horizon region are fixed by the attractor mechanism, they can be suppressed as above and the central charge can be computed from just the Chern-Simons supergravity. According to the construction in section 4, the SU (1, 1|2)L × SU (1, 1|2)R Chern-Simons theory will induce an algebra with eight supercharges in each factor, so that the boundary theory will be a (4, 4) superconformal one. The right and left algebras are then the ’small’ N = 4 algebra, which contains only one SU (2) R-symmetry group that rotates the supercharges in doublets. This follows from the same property of the eight supercharges of the SU (1, 1|2) supergroups in the AdS3 theory. This is exactly the superconformal algebra underlying the microscopic CFT on the D1-D5 intersection23 [27]. A property of this N = 4 algebra is that the level of the two SU (2) currents must be equal to 1/6 of the central charge. As this level was computed in (8.28), it follows that the central charge in this case must be c = 6k = 6N1 N5 . We see that the final result is the same central charge as the naive Brown-Henneaux one, used above. 23 For a list of the superconformal algebras induced by various AdS3 supergroups, see [36] 60 8.1.3 Relation with anomalies The presence of anomalies in the microscopic CFT can be used both as a consistency check for the AdS/CFT point of view taken here and as a more general argument in favor of the approach taken in this section. Start with the microscopic description in terms of a 1 + 1 dimensional theory which flows in the IR to a (4, 4) superconformal theory. The worldvolume theory contains fermions transforming chiraly under the SU (2)L × SU (2)R R-symmetry, so that it is subject to an anomaly. This anomaly coefficient is found to be N1 N5 [27] and is the same for all energies away from the conformal point, as it is an invariant property of the theory. Furthermore, at the conformal point it is related to two other quantities: it is equal to the level of the two SU (2) current algebras associated to the R-symmetry, which in turn is 1/6 of the central charge by imposing the (4, 4) superconformal algebra. All these being known, the AdS/CFT dual theory of the near horizon AdS3 should reproduce this in some way. As first noted in [51], the anomaly on the boundary theory should descent from a Chern-Simons term in the AdS bulk. This is simply the statement that any anomaly associated with a symmetry on the boundary theory is by definition a failure of the effective action to stay invariant under gauge transformations. As the partition function of the bulk and boundary theories are to be equated, this means that the bulk action should also vary in a way that reproduces the variation of the boundary action. The only such term is the three dimensional Chern-Simons term suggested by anomaly descent on a two dimensional theory. This term is constructed in the framework of the Stora-Zumino approach to the anomaly so as to produce the anomalous variation of the two dimensional action. In the present context, a Chern-Simons term in the bulk is furthermore consistent with gauge invariance of the bulk equations of motion, as it transforms into a boundary term: Z Z k 2 k ICS = T r A ∧ dA + A ∧ A ∧ A ⇒ δICS = T r(vF ). (8.29) 2π M3 3 2π ∂M3 Since the bulk gauge fields couple to the conserved currents of the boundary theory, the boundary values of the gauge field can be identified with a two dimensional gauge field associated with the symmetry expressed by the currents. When the partition functions are set to be equal, the last equation becomes the variation of the two dimensional theory, or in other words the anomaly. In the case of the D1-D5 system studied here, the coefficient of the Chern-Simons terms was indeed found to be equal to N1 N5 , which matches exactly the anomaly of the boundary theory. Of course, this is in a way trivial as the boundary theory is superconformal, so that the anomaly, the current algebra level and the central charge are all essentially the same quantity and the anomaly point of view is not different from the one adopted before. What makes anomalies interesting is that one can argue about the exactness of the results found here in the full string theory including higher derivatives. The basic idea relies on the fact that the full sting theory should be anomaly free. Therefore, in all cases, any noncanceled anomalies of the worldvolume theory on a system of branes 61 should in fact be cancelled by coupling it to the bulk supergravity theory through some current flowing from the bulk to the worldvolume that cancels the anomaly. This is the so-called anomaly inflow mechanism. In the present case, it is possible to cancel the worldvolume anomalies [53], [54]. The total anomaly on the d-dimensional theory such as on an intersection of D-branes is expressed as a d + 2 form, called the anomaly polynomial. This is made up of certain combinations of characteristic classes of the tangent and normal bundles of the intersection. For example, in the D1-D5 system, where the intersection is 1+1 dimensional24 , this would be in principle some 4-form written as a combination of lower degree forms taking values on its tangent bundle and on its SO(4) normal bundle. But it was shown in [53] that this form must necessarily include a factor of the Euler class χ of the normal bundle N of the intersection. Now, in the case of the D1-D5 system it happens that the χ(N ) is already a four form, so that the anomaly polynomial must be proportional to it. The exact expression is given by: P4 = 2πN1 N5 χ(N ) = N1 N5 F ∧ F, 16π (8.30) where F = dA + A ∧ A is the field strength for the SO(4) normal bundle connection A defined on the base space (the intersection of the branes). The two dimensional anomaly is given by anomaly descent in the usual way as: P2 = 2πN1 N5 χ(1) . (8.31) Here we use the anomaly descent notation χ(N ) = dχ(0) , δχ(0) = dχ(1) , where the forms χ(0) , χ(1) are the Chern-Simons form and its variation in (8.29). This is what has to be cancelled by a contribution of the bulk supergravity theory. The way to do this is the following. Since the D1 and the D5 branes are the electric and magnetic objects coupling to the RR two form Bµν , it is sufficient to consider just that field instead of the full Type IIB supergravity. It is convenient to work in six dimensions, where the intersection of the branes becomes a string carrying both electric and magnetic charge under B. As we are dealing with a dyonic object, it is not trivial to construct gauge fields for the three form field strength H. A solution was put forward in [54]. Consider the four form J, which is defined to be a δ-function supported Poincaré dual form for the dyonic string. This means that it is of the form J ≈ δ(y)dy 1 ∧ . . . dy 4 for appropriately chosen coordinates y 1 . . . y 4 centered on the string25 . The Bianchi identities and equations of motion become: 1 1 dH = d ? H = J, g5 g1 (8.32) where g1 , g5 are the 1- and 5- brane charges. The Bianchi identity can be solved by the ansatz: H = dB + g5 K, 24 K = χ(0) + e3 , (8.33) We are supressing the compact M4 manifold throughout, so that the discussion is actually done in six dimensions. The extra four dimensions can be trivially added as the D5 brane wraps them, and will not be part of the transverse space considered here. 25 To write its explicit form a precise definition of the delta function must be given, but this will not be needed in the following. 62 where χ(0) is the Chern-Simons form defined by χ(N ) = dχ(0) and e3 is the SO(4) invariant three form polynomial shown in (8.21). This combination of χ(0) and e3 is constructed to give the current J above when acted upon with the differential operator. Nevertheless, this definition is ambiguous due to the noninvariance of χ(0) under diffeomorphisms of the normal bundle. This is expressed as a variation of K by an exact three form under SO(4) transformations, forcing the two form potential to vary as well: K → K + dQ B → B − g5 Q, (8.34) if (8.33) is to kept invariant. The two from Q is arbitrary in the bulk, but is constrained to reduce to the gauge variation of the Chern-Simons form χ(0) on the intersection I: dQI = δχ(0) ≡ dχ(1) QI = χ(1) , (8.35) where the two form χ(1) appeared as in the descent formalism above. Now, it is easy to put all this together and see that adding to the action a term of the form: Z (8.36) Sinf = g1 B, I the anomaly (8.30) will be exactly cancelled upon variation. Indeed, by (8.34) the variation of this term under a normal bundle diffeomorphism is: g1 δB I = −g1 g5 QI = −g1 g5 χ(1) . (8.37) By use of the Dirac quantisation condition for the charges, this exactly cancels the worldvolume anomaly in (8.31). We thus see that the full theory is nonanomalous if the term (8.36) is added to the supergravity action. Going to the gravity dual of this situation, the intersecting branes are replaced by a near horizon AdS3 space coupled to the same supergravity theory on the ’outer’ asymptotic flat region. As this is still the same string theory, the only difference being the larger gravitational coupling, the anomaly inflow from the ’outer’ supergravity theory must be the same, as this is coupling independent. Invoking again the anomaly cancellation of the whole theory, it follows that the same current should flow outwards from the near horizon AdS3 region as it did from the worldvolume of the branes, as in this picture the boundary of the AdS3 space is just a Gauss surface through which the total flow of the anomalous current must be zero. This shows that the total anomalies descending from the bulk to the boundary should in fact be exactly the same as the anomalies on the worldvolume theory on the branes. In the case of the D1-D5 system at hand, the total anomaly descending on the boundary comes from the Chern-Simons terms found above, with the expected result. Note that in the AdS3 supergravity the extra term (8.36) does not contribute to the boundary anomaly, as its variation (8.34) under normal bundle diffeomorphisms is zero at the AdS3 boundary. This is because the two form B appears in the action integrated over the source at the center of the AdS3 space, but is otherwise arbitrary in the bulk. We conclude that the full anomaly is controlled by the Chern-Simons terms. The overall conclusion from a purely supergravity point of view is that the Chern-Simons terms in the near horizon AdS3 region uniquely determine the anomalies of the boundary 63 theory, which in turn give the exact result for the central charge of the theory at the conformal point. Of course, this supposes that one knows all of the Chern-Simons terms arising in an AdS3 compactification of a higher dimensional supergravity, a far from trivial task. In the D1-D5 case considered here, the relevant six dimensional (2, 0) supergravity on the AdS3 × S 3 background has been heavily studied and the correct terms are known, at least for the two derivative action. Finally, a comment on the higher derivative corrections contained in the full Type IIB string theory effective action. A full treatment of the D1-D5 system should take into account these terms as well and verify that their contribution does not alter the above central charge, which is known to be the correct one from the microscopic calculation. But if this effective action is to reproduce the low energy string theory dynamics, it should be anomaly free. In particular, it should lead to the same anomaly as above, which is exactly known from the microscopic theory. Thus, if we insist on the total anomaly cancellation of the full string theory, any other contributions to the Chern-Simons terms from higher derivative terms are forced to cancel, leaving the same result for the anomaly. Using the constraints of N = 4 supersymmetry, this kind of reasoning leads to the same result for the central charge as well. This argument for the exactness of the central charge is the reason for introducing the anomaly point of view here. 8.2 M 5 brane on a Calabi Yau manifold The procedure taken for the D1-D5 system can also be followed for the case of a black hole solution in a four dimensional M-theory compactification. The compact manifold is taken to be CY3 × S 1 on which the M 5 and M 2 branes are wrapped in the way described in section 7.3. Considering the decoupling limit of the metric in (7.21) by keeping only the two dominant terms in Z I and k, one finds the following expression: dr2 r ds2 = 2 dtdz + Cdz 2 + R2 2 + R2 dΩ22 . R r (8.38) Here, the metric for the (constant volume) CY manifold is omitted, so that this is effectively the near horizon of a black string solution of the N = 1 supergravity in five dimensions. The constants appearing are 1 1 R = ( CIJK pI pJ pK )1/3 2 6 1 RC = 2q0 + DAB qA qB . 6 (8.39) Note that we again suppress the values of the gauge field strengths that support the sphere. They will become important in the next subsection. The metric in (8.38) is a product of a locally AdS3 space with the two sphere, so that we find again an example of the AdS3 /CFT2 correspondence, since the microscopic system is a 1 + 1 theory on the M 5-brane worldvolume. As in the D1-D5 system, the CY volume is constant in the large charge limit. On the contrary, both the AdS3 and the sphere radii scale with the charges, justifying the omission of the CY from the near horizon considerations. Taking a closer look at the near horizon and microscopic theories, a matching of the relevant 64 symmetries is found again. The spacetime symmetries match in the usual way and in both cases there is a SO(3) ∼ = SU (2) symmetry, one from the S 2 fibration of AdS3 and one from rotations of the normal bundle of the five dimensional string in the microscopic description. These rotations give rise to the SU (2) R-symmetry rotations of the small N = 4 algebra on the worldvolume (0,4) theory on the string. To generalise this to the full supersymmetry algebras of the two theories one must examine the structure of the near horizon AdS3 supergravity. When discussing this limit of the solution, it is important that the reduction of five dimensional supergravity on AdS3 × S 2 is again a maximally supersymmetric one. This can be justified using the same line of thought as in Eqs. (8.15)-(8.16), by replacing the supersymmetry variation of (8.15) by: i I N P P (8.40) δψM = DM (ω)ξ + XI (ΓN M − 4δM Γ )FN P ξ. 24 Here, XI are the scalars from the compactification of the eleven dimensional theory (see below), which are constant in the near horizon area. By the same procedure as in (8.16), it is shown that the background (8.38), (7.25), (7.26) preserves all the supersymmetries of the original theory26 . It then follows that the decoupling limit of the solution takes the form of an AdS3 ×S 2 Freud-Rubin solution asymptotically, which is all that is needed for the AdS/CFT correspondence to work. The discussion concerning the reduction of the supercharges on AdS3 in subsection 8.1.2 applies here as well. The only difference is that the supersymmetry parameter is now a single complex four component spinor in five dimensions which can be viewed again as spanning a two dimensional symplectic space. By reducing this in a 2 ⊗ 2 way on the AdS3 and the two sphere, one gets four real spinors in three dimensions transforming as two doublets under SO(3) ∼ = SU (2) rotations on the sphere, exactly as in subsection 8.1.2. Now recall that all of the supercharges were chosen to transform under the representation with Γ11 = 1 in eleven dimensions (see section 7.3). A similar property persists in the five and three dimensional reductions, namely one gets γ̃ 6 = γ 3 = 1 (tilde denotes spinor representations in five dimensions). This means that the three dimensional supercharges should be part of a definite (right or left) supergroup, since the right and left supergroups contain supercharges in one of the representations with γ 3 = ±1. By the result in section 7.3 that the supercharges preserved by the black hole solution are rightmoving one is led again to the unique choice for the right supergroup in the three dimensional Chern-Simons supergravity that contains the bosonic SO(2, 2) × SU (2): SU (1, 1|2). In the same way explained for the D1-D5 case, this AdS3 supergroup leads to a rightmoving small N = 4 superconformal algebra. We have thus recovered a dual (0, 4) CFT, in agreement with the AdS/CFT correspondence. What has to be determined now is its central charge and the asymptotic L0 charges. The asymptotic charges of the locally AdS3 space in (8.38) can be computed easily. Since we are considering the extremal case, the resulting metric is expected to be that of an extremal BTZ black hole. Upon the change of coordinates Rz Rz R 2 CR t= u + Rz φ, z = − u + Rz φ r= ρ − , (8.41) 2R 2R 2Rz2 2 26 Note that the lower index here is defined as XI = 16 CIJK X J X K 65 with Rz the radius of he coordinate z along the circle, (8.38) becomes the following BTZ metric with radius l = 2R: ds2 = − (ρ2 − CRz2 )2 2 CRz2 4R2 ρ2 dρ2 2 2 du + ρ (dφ − du) + + R2 dΩ22 . 4R2 ρ2 2Rρ2 (ρ2 − CRz2 )2 (8.42) Thus, the mass and angular momentum of the BTZ black hole are read off to be: Ml = J = CRz2 . 8G3 R (8.43) The last parameter we need is G3 = G4 Rz /2R2 , as found by the same method as in (8.4). Using the Brown-Henneaux central charge with the radius of AdS3 of the metric above, the Cardy formula for the excitations of the boundary CFT gives the entropy: s r c DAB qA qB c CIJK pI pJ pK S = 2π (L0 − ) = 2π q0 + , (8.44) 6 24 6 12 which is the same as (7.19) for large charges and the appropriate convention for G4 = 1/4. Note that the difference resides only in the central charge term, which is modified by the higher derivative corrections. As in the D1-D5 case, we now proceed to find the relevant Kaluza-Klein and gravitational Chern-Simons terms in the AdS3 action. Unlike that case however, this will give corrections to the central charge and the final result for the entropy will be exactly as in (7.19). 8.2.1 Compactification on S 2 In order to find the relevant Chern-Simons terms in the three dimensional action in this case, the action in the higher dimensional case is needed. In view of the discussion in subsection 8.1.3, it is crucial to use a version of the action that guarantees the anomaly cancellation on the branes involved. In the D1-D5 case the extra term was actually irrelevant in the end, but this is by no means the general case. In the present context we need the terms in eleven dimensional supergravity that are responsible for anomaly cancellation on the M 5 brane by anomaly inflow. This involves adding a Chern-Simons term of fourth order in the Riemann tensor to the standard bosonic part of the eleven dimensional supergravity action in order to cancel the tangent bundle anomaly on the M 5 brane [12]. The normal bundle anomaly is conveniently cancelled by the Chern-Simons term for the three form gauge field already present in the two derivative action. The action then takes the form: Z Z √ 1 1 2 1 11 −g(R + F(4) ) − I=− 2 dx A(3) ∧ F(4) ∧ F(4) 2 2k11 2 12k11 Z 1 1 4 2 2 − A(3) ∧ T rR − (T rR ) . (8.45) 2 1/3 3 · 26 · (2π)10/3 (2k11 ) 4 In this and following expressions, ki2 = 8πGi and the Riemann tensor is viewed as a two form taking values in T 1,1 (M11 ), on which space the trace is defined. Note that the Chern-Simons 66 terms shown are the only ones allowed by gauge invariance and that their coefficients are protected from higher loop corrections (the last term is a one loop correction). As the main idea behind the arguments in subsection 8.1.3 relies on the anomaly inflow mechanism, there should be no corrections to the final result for the central charge found by considering just these terms. This follows from the fact that any other higher derivative corrections to the action should leave the anomaly inflow to the M 5 brane unchanged if the whole theory is to be consistent. In other words, even though we are not excluding the presence of these higher derivative terms, we rely on the requirement of the eventual cancellation of their contributions in this particular case that involves a quantity (the anomaly) that is exactly known and consistently reproduced by the terms kept. The next step is reducing this action on a CY manifold to get the bosonic part of five dimensional N = 1 supergravity. This means expanding the four form field strength on a P basis of harmonic two forms on the CY as F(4) = I F I ωI , giving a set of abelian gauge fields in five dimensions labelled by I. The components with all four legs in either the CY or in the five dimensional space give extra scalars which are suppressed. Through the same expansion, the last term in (8.45) will give some terms with precisely two of the four factors of the Riemann tensor having both of their form indices on the CY. 1 2 These terms then will be proportional to the second Chern class c2 = (2π) 2 R of the CY tangent bundle. Since this is a harmonic four form, it can be expanded into the basis of harmonic four forms Poincaré dual to the one the four form field strength was expanded: c2 = c2I ω I . R The upper index here denotes this duality, as ω I is such that ω I ωJ = δJI on the CY. The final five dimensional action is of the form27 : Z Z √ 1 1 1 5 I Jµν CIJK AI ∧ F J ∧ F K I = − 2 d x −g(R + GIJ Fµν F ) − 2k5 4 12k52 Z c2I 2 + F I ∧ T r(ΓdΓ + Γ3 ). (8.46) 2 384π 3 Here, CIJK are the intersection numbers of the CY as before, whereas GIJ is a matrix depending on the topological properties of the CY and its moduli that will not enter the following discussion. Note that the moduli have been frozen to constant values in this action. This is because they are constrained by the attractor mechanism to take specified values in the near horizon region of the black hole backgrounds considered here. Furthermore, any scalars in the AdS3 action are irrelevant to the computation of the central charge of the boundary CFT. We are therefore allowed to drop them from the outset. The final step is the reduction of this five dimensional action to the AdS3 ×S 2 near horizon geometry (8.38) of the black string. Similar to the D1-D5 case we are after the Chern-Simons terms for the SO(3) Kaluza-Klein gauge fields coming from the S 2 reduction and the possible gravitational Chern-Simons terms. As the interesting part of the geometry is its boundary, 27 We omit a number of extra terms in which this structure of indices in the last terms does not occur. These terms involve extra fields from the compactification (scalars, three forms etc.) which do not lead to Chern-Simons terms and will play no role in what follows. 67 we proceed in performing a Kaluza-Klein reduction near that region. This allows us to use the asymptotic form of the gauge field strengths near the boundary shown in (7.25), thus justifying a Freud-Rubin ansatz for the AdS3 × S 2 , as in the D1-D5 case. As done previously, the ansatz includes only the metric and the gauge fields in a simplified setting that is enough for the present purposes. A more general discussion, which also involves the dilaton, can be found in [59]. The procedure is indeed very similar to the D1-D5 case, as can be seen from the following ansatz: ds25 = ds23 (AdS3 ) + R2 δij Dxi Dxj , Dxi = dxi − AiKl xl pI F I = −2πpI e2 ≡ − ijk Dxi Dxj xk − FKij xk , (8.47) 4 P i i where i, j, · · · = 1, 2, 3, and x x = 1 and we use a subscript K to distinguish the KaluzaKlein field strength from the five dimensional ones. Note the implicit definition of e2 , which is completely analogous to e3 in the five dimensional case. The properties of e2 are that it integrates to unity on the sphere, its gauge invariance and closeness, as well as its invertibility with respect to the differential operator. This makes it an ideal candidate for an ansatz for (0) the field strength. Upon defining e2 = de1 , the triple intersection term in (8.46) becomes: Z Z 1 2π 3 (0) I J K I J K ST I = − C A ∧ F ∧ F = 2 CIJK p p p e1 ∧ e2 ∧ e2 . (8.48) 2 IJK 12k5 3k5 When reducing this to an AdS3 integral, it is not hard to see from (8.47) that if the part of e2 with both legs on the sphere is chosen, then the remaining integrant on AdS3 must be of the (0) form AK ∧ FK , because the form e1 must contain an explicit gauge field. The final answer is provided by a result of Bott and Cattaneo [55]: Z Z 1 2 3 (0) e1 ∧ e2 ∧ e2 = − 2 T r AK dAK + AK . (8.49) 8π 3 Plugging this in (8.48), it becomes: ST I CIJK pI pJ pK =− 24π Z 2 3 T r AK dAK + AK , 3 (8.50) where we switched to conventions such that k52 = 2π 2 , so as to match with (7.19). This is a right moving Chern-Simons term with 6k = CIJK pI pJ pK , that contributes to the right moving central charge c of the boundary CFT, by the same argument as for the D1-D5 case, since the right moving sector is a N = 4 superconformal CFT. The second Chern-Simons term in (8.46) will give an extra contribution to this same Chern-Simons term for the Kaluza-Klein gauge field and (as is obvious) an extra gravitational Chern-Simons term. This comes about by just substituting the ansatz for F I in (8.47) and making use of the definition of F and the fact that the integral of e2 over the sphere is unity. The result is: Z c2I pI 2 3 ΓdΓ + Γ , (8.51) Sgr = 192π 3 68 where the Christoffell connection is defined on AdS3 but has still five dimensional tangent space indices. This means that it has to be decomposed in objects with three dimensional indices. Upon use of the ansatz in (8.47), the forms Γiµj with two indices on the sphere are found to be equal to the corresponding Kaluza-Klein gauge potentials. When changing to standard SU (2) notation (used in the reduction of the previous term), we get AK = 2i ijk σi Aij . The other terms either give Christoffell connections on AdS3 or vanish. The final result a sum of two terms: Z Z c2I pI c2I pI 2 3 2 3 (8.52) Sgr = − AK dAK + AK + ΓdΓ + Γ . 48π 3 192π 3 This gives us an extra contribution to the Chern-Simons term of the Kaluza-Klein gauge fields and a gravitational Chern-Simons term. The first one shifts the result for the right moving central charge by 21 c2 · pI . The second one gives the difference of the right (c̃) and left (c) moving central charges according to the results of subsection 4.4 as: c − c̃ = 12 c2 · p. The final result for the central charges is: 1 c̃ = CIJK pI pJ pK + c2 · p, 2 c = CIJK pI pJ pK + c2 · p. (8.53) These are exactly the central charges found from the microscopic side in [9]. By use of this central charge in 8.44 instead of the Brown-Henneaux result, one gets presicely the microscopic result in (7.19). The discussion of anomalies of the D1-D5 system is of course applicable in the present context. In exactly the same way, the right moving anomaly descents from the Chern-Simons terms for the Kaluza-Klein gauge fields. The sole difference is that only the right moving sector is supersymmetric, which means that only the corresponding central charge can be found by gauge anomaly arguments. The left one was found by tangent bundle anomaly arguments in the same spirit. Again, because the anomalies are the same for all values of the coupling and independent of possible loop or string corrections, the same arguments of exactness of the final result apply in this case as well. A final important point in this construction is the following. What was just shown here is that the full result for the central charges of the boundary CFT can be found by consideration of the Chern-Simons terms, using the fact that the solution of the lowest order action gives rise to a near horizon AdS3 region. Now, even though the assumption that this feature persists when higher derivative corrections are included is justified on physical grounds and has been verified recently in [8], there is no general argument that can reliably give the eigenvalue of the L0 operator. This has to be found independently, by an explicit construction of the near horizon geometry of the system in question. In particular, this means that a computation of the entropy including higher derivatives should not in fact use the L0 eigenvalue implied by (8.42), as this is the solution to the tree level action. Since this is an asymptotically flat solution, the charges do not get any modifications from the extra five dimensional ChernSimons term due to the boundary conditions, but the functional form of the L0 eigenvalue could in principle change. A proper treatment would follow the lines of [8], where this kind of analysis was carried out and full agreement was found for the case of zero electric charges. 69 In the more general case of adding electric charges, one would still expect the asymptotic form of the AdS3 solution near its boundary to be of a form similar to the one presented here, with the magnetic parts of the field strengths dominating over the electric ones, so that the situation would be reduced locally to that in [8]. At any rate, the example given in this section reinforces the consistency of the whole construction as the exact microscopic result is matched from the macroscopic side in a nontrivial, chiral example. 8.2.2 Black rings Although so far we have only discussed examples where a black hole emerges as the dimensional reduction of a black string, the validity of the approach taken in previous sections extends to any case where there is an AdS3 factor in the decoupling limit - that is when the microscopic theory is a two dimensional one. A manifestation of this is shown in this next subsection, where we apply previous results to the black ring in five dimensions. As discussed in section 7.4, the case of the black ring in five dimensions can be seen as a different phase of the same microscopic system as the four dimensional black hole. The only difference is the fact that in order to go to the black hole case one has to dimensionally reduce along the five dimensional string. As the microscopic CFT describing the two systems is the same, it should be generally expected that the near horizon region of the black ring is that of AdS3 × S 2 , with the two factors having the same length scales as in (8.38). Indeed, this is the case if one considers the near horizon region of the black ring metric (7.34) by setting r = −R/y and taking the limit −y >> 1. After a charge of coordinates of the form: A2 A1 B1 B1 0 0 dt = dv − + + A0 dr, dφ = dφ − + B0 dr, dψ = dψ − + B0 dr, (8.54) r2 r r r where the constants Ai , Bi are determined so that the final solution is smooth and can be found in [57], the metric becomes: ds2 = 2dvdr + p2 2 4Lr dvdψ 0 + L2 ψ 0 2 + dθ + sin2 θ(dφ0 − dψ 0 )2 . p 4 Here, a rescaling r → Lr/R was done, and we have set p = (p1 p2 p3 )1/3 , cos θ = x and: s 1 2 3 ppp 1 IJ 1 1 2 3 − Jψ + D qI qJ . L= 2 p p p p 4 12 (8.55) (8.56) As expected, this metric is that of an extremal BTZ black hole times the two sphere with the length scales of the similar metric (8.38). The values of L0 , L̃0 are found in the usual way by the mass and angular momentum of this BTZ solution: lM = J = 70 L2 . 4G3 l (8.57) Using the facts that l = p and G3 = G5 /πp2 = 1/4p2 , one easily finds the entropy by the Cardy formula: r c c S = 2π (L0 − ) = 2πp2 L. (8.58) 6 24 Here the Brown-Henneaux central charge was used, since we are considering the case of the Calabi-Yau manifold being the six torus, which means that c2 = 0 identically. This result is exactly (7.39) if (8.56) is used. As the microscopic theory is the same, it was to be expected that the central charge turns out to be the same as for the four dimensional solution (without the c2 correction). In fact, the decoupling limit of the geometry is asymptotically AdS3 × S 2 supported by flux of the gauge fields on the sphere, in exactly the same way as for the previous case. This can be verified by taking the decoupling limit of the expression in (7.37) and then considering its asymptotic behaviour, which is of the form (7.25). Of course, the mass and angular momentum of the BTZ black hole are not the same as in the previous example. This is what distinguishes the black ring phase of the system from the four dimensional black hole in this approach. Nevertheless, note that the form of the final L0 eigenvalue is of the same form as in the four dimensional case, as the momentum charge q0 is interpreted as angular momentum Jψ along the ring and there is also a term of the form DAB qA qB . The extra term proportional to the central charge is a subleading term in the four dimensional case [58]. Moreover, this is consistent with the proposed connection between four and five dimensional solutions in [34]. 9 Discussion and outlook In the present thesis, an exposition of the AdS3 based approach to the computation of black hole entropy was given [5, 6]. We devote this last section to a discussion of the merits and limitations of this point of view. The most profound feature of this approach is that it is heavily based on the AdS/CFT correspondence between a 1 + 1 dimensional CFT on the microscopic side and a locally AdS3 near horizon region on the macroscopic side. This duality preserves the highly constrained nature of the microscopic CFT’s, which in all cases treated here are N = 4 supersymmetric nonlinear sigma models, allowing a rather formal treatment with no refence to their detailed structure28 . The dual picture is that of a supergravity theory with no local degrees of freedom in the bulk, equivalent to an induced theory at the AdS3 boundary. We have indeed found an intricate interplay between these rather special properties of the bulk AdS3 theory and is dual. The final result of this line of thought gives an exact match of the entropy computed through the AdS3 theory with both the microscopic and the previously known macroscopic result. This is required if the AdS/CFT correspondence is valid. As was explicitly seen in section 4, the few ingredients needed in a saddle point evaluation 28 Due to an identical constraint relating the central charge to the level of the R-symmetry algebra, this line of thought can be applied to N = 2 theories as well 71 of the entropy in the microscopic theory, namely the central charge and the L0 , L̃0 eigenvalue, are mapped to equally simple quantities in the dual AdS3 theory, namely the coefficient of the Chern-Simons terms and the mass and angular momentum. This was reinforced later through the anomaly argument of subsection 8.1.3, which leads to a claim of exactness of the result for the central charge even if higher derivative terms are added. The computation of the entropy is then made a simple matter after the setup in section 6 is considered, provided that the coefficients of the Chern-Simons terms can be found and an explicit solution is known so that the mass and angular momentum can be extracted. This raises the following two subtleties of the whole approach. First, one must be able to find all the relevant Chern-Simons terms in the reduced AdS3 action if the entropy match is to have any chance of being correct. Even though this might look like a messy job, the anomaly argument provides a general guide to choose the terms that should be considered in the higher dimensional setting, as was done for example in the case of the wrapped M 5 brane. As the subject of anomaly cancellation has been studied in depth, one can use the results in the literature to find the relevant terms to be added in the action, relying on the fact that all other contributions should in fact cancel themselves. In any case, treating even this limited number of terms is not empty of tricky points, as was seen above. Furthermore, an actual check of the cancellation of the other contributions at some order in the gs and α0 expansions would be an interesting but highly nontrivial task. The second point is the limitation of the approach to the cases where an explicit solution is known, due to the need to find the mass and angular momentum of the locally AdS3 space. This means that even though one can argue his way out of using the full action when computing the central charge, an actual solution to the action used has to be found (at least in the near-horizon region). This is not trivial, given the fact that if the solutions are to preserve some supersymmetry, the supersymmetric completion of the terms used for anomaly cancellation should be used. This was tried recently in [8] for the case of the wrapped M 5 brane and agreement with the microscopic result was found. Here, we used the values for the mass implied by the solution of the tree level action, using the fact that the charges it depends on do not receive corrections from the extra Chern-Simons term in five dimensions. Nevertheless, the functional form of this dependence could be changed, even though this is unlikely and was actually checked not to happen in [8]. On the other hand, this dependence of the scheme on the explicit solution gives an extra freedom. As seen in the case of the black ring, the central charge is the same as in the case of the four dimensional black hole. This is in accord with the fact that the microscopic picture is the same, but the L0 eigenvalue is different, leading to the correct entropy. This means that the partition function defined in the AdS3 space through the procedure in section 6 is general enough to accommodate for different phases of the same microscopic system. The obvious overall disadvantage of the approach is its restriction to cases where the microscopic low energy theory on the D-brane worldvolume is accurately described by a 1 + 1 dimensional CFT on an effective string. This means that the standard AdS2 × S p near horizon geometry of an extremal black hole in p + 2 noncompact dimensions can be combined with the extra dimension along the string, and get uplifted to an AdS3 × S p geometry. As was seen 72 in the explicit examples, this larger geometry naturally encompasses the near extremal case since it is closely related to the microscopic picture through the AdS/CFT correspondence. This is certainly not always the case, as the mere existence of an effective string encompassing all the low energy degrees of freedom means that the microscopic description in terms of Dbranes allows for a distinguished one dimensional cycle in the compact space. As the entropy arises mainly from light open strings stretched between the D-branes, this also means that quantities along this distinguished cycle must control the possible ways of arranging the Dbranes, the so called moduli space of the system. In the case of the D1-D5 system this is automatic, in the sense that the moduli of the circle along the D1-branes are all the moduli of the system because the D5-branes wrap the whole of the internal space. In the wrapped M 5 case, the situation is more complicated, but it is still true that the moduli of the M 5 and M 2 branes can be described by scalars in a reduced theory on the M-circle [9]. This nice picture does not extend to cases such as the BMPV black hole when it is embedded in five dimensional N = 1 supergravity. This can be viewed in M-theory as a compactification of eleven dimensional supergravity on a Calabi-Yau manifold with a set of M 2 branes wrapping its two-cycles. In this picture all branes are on an equal footing, and moreover there is no U (1) isometry on a general Calabi-Yau that could be picked out to serve as a distinguished cycle. It therefore follows that the approach described here can not be applied. Other examples where no such circle exists include so-called small black holes arising from fundamental heterotic strings in dimensions higher than five (see [61] for a related discussion). Nevertheless, it was recently shown [33] that the BMPV black hole develops a factor of AdS3 at its near horizon region close to the maximally rotating limit (when the entropy in (7.18) tends to zero). We end this discussion with a brief account of further extensions of the approach and open problems. In the examples shown here, where the AdS3 based approach is applicable, the partition function defined in section 6 can actually be calculated by summing over all possible geometries, by starting with the regular ones and applying SL(2, Z) transformations to get black hole geometries. This is related to the expansion in [31], where a similar split of states was considered. A quantity of interest computed in this way is the generating function for the degeneracies of BPS states at any given charge, called the elliptic genus. This is computed on the basis of the formalism set up here and compared with known results in [5], [7]. These extensions open the way for entropy computations beyond the saddle point approximation, to include contributions which are subleading in the charges. Finally, an interesting connection is with the OSV conjecture [10] for four dimensional black holes, which relates the above degeneracies to the topological string. 73 A The decoupling limit Here, we take the time to carefully define Maldacena’s decoupling limit [26]. This is essential to the discussion of the near horizon geometry of four and five dimensional black holes. When these black holes are embedded in string theory as a system of intersecting Dbranes, the decoupling limit gives us a gravity/gauge theory duality that is known to contain the right degrees of freedom. This limit is defined as29 : r ls , r → 0, = const. (A.1) ls2 Here, r refers to the distance from the center of the geometry (the horizon) and ls is the string length. Actually, there can be more parameters in the problem in need of a definition of their scaling properties under this operation, but they will be introduced below. What is important is that this limit keeps the energy of the near horizon excitations finite while rising arbitrarily the gravitational potential between the horizon region and the asymptotic flat region. These properties can be verified by consideration of the p-brane supergravity solutions of section 2. This isolates the near horizon dynamics and makes possible a comparison with the theory on the D-brane worldvolume, which is also decoupled from the ambient flat space in this limit. To treat a few details, one of the parameters in the cases discussed here is the volume of the internal compact manifolds which are always present in black hole constructions. It is defined to scale as a constant in length units of ls : V /lsn = const., where n is the manifold’s dimension. With this definition, the part of the metric describing the compact manifolds will always scale as ls2 ∼ α0 . Next, we turn to the behaviour of the various factors appearing in the metrics of black holes built up from p-branes, emphasizing the examples treated here. Starting with the D1-D5 system, consider (7.7) combined with (7.8). Using the scaling property for the compact four manifold just defined, it follows that G5 scales as ls4 ∼ (α0 )2 . This means that both of the Q1 and 0 2 0 Q5 terms scale as rα2 , which blows up. On the other hand, the QP term scales as αr , which is to be held constant. This means that in taking the decoupling limit one ’drops the one’ from the Hi factors attached to p-branes but not from the ones corresponding to momentum excitations. In the case of wrapped M-branes, the same rules apply as one can easily verify using the fact that the constants in the Hi factors are just wrapping numbers which do not vary. Next, consider the scaling behaviour of the nonextremality parameter µ in the function in (2.12), which is defined to scale as µ ∼ (α0 )2 . It follows that the factors of µ/r2 are to be held constant, which is consistent with the conclusion in the previous paragraph that the constants in the factors associated with momentum are not to be dropped. Finally, the rotation parameters l1,2 in (7.15) scale as α0 . This means that the constants in (7.16) have to be dropped. These scaling properties for the various quantities lead to the important conclusion that the decoupling limit of a black hole is a near extremal limit. This is clear from (2.15), by imposing that the charges related to D-branes scale like α0 as above. This leads to sinh 2ai ∼ 1/α0 → ∞, 29 In section 2 the string tension α0 = 2πls2 was used instead. 74 which (combined with the vanishing µ) is exactly the extremal limit defined below Eq. (2.16). For the momentum charge the situation is different, as one should have QP ∼ (α0 )2 . In this case (2.15) leaves ap finite but not unrestricted, as all charges must be large for the supergravity approximation to be valid. One then has the clearly near extremal situation 1 aP ai → ∞, which is the so-called dilute gas regime. In this regime the interactions between left and right moving excitations on the effective string in six dimensions can be neglected and a reliable microscopic calculation can be done [60]. Using the scaling rules for the various quantities above, one can derive the near horizon form of the various metrics considered in this thesis. One final point is the appearance of an overall factor of α0 in the near horizon metric which is consistently omitted. B Formulae used in the S 3 reduction Here, the explicit formulae for various forms used in the main text for the S 3 reduction are stated and briefly motivated. The metric on S 3 used in the text is: (B.1) ds2 = dx21 + dx22 + dx23 + dx24 (xi )2 =1 = dθ2 + cos2 θdψ 2 + sin2 θdφ2 . The SU (2) × SU (2) invariance of the metric is seen in complex coordinates: i z1 = cos θ/2e 2 (φ+ψ) , i z2 = cos θ/2e 2 (φ−ψ) ⇒ ds2 = dz1 dz̄1 + dz2 dz̄2 . (B.2) In these coordinates, (z1 , z2 ) and (z1 , z̄2 ) both transform independently under SU (2)L and SU (2)R respectively. This is related to the isomorphism SO(4) ∼ = SU (2) × SU (2) of the isometry group of the three sphere. The SO(4) Lie algebra [J ij , J kl ] = δ ik J jl + δ jl J ik − δ il J jk − δ jk J il , J ij = −J ji , i, j, . . . = 1, . . . , 4 (B.3) can be decomposed as two SU (2) algebras if: JLa = − 1 4aij ij J + J 4a , 2 JRa = − 1 4aij ij J − J 4a , 2 a = 1, 2, 3 (B.4) For these linear combinations, the SO(4) algebra becomes a sum of two standard SU (2) algebras for the right and left operators. 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