Asymptotically-flat supergravity solutions deep inside the black-hole regime

Journal of High Energy Physics, Feb 2018

Iosif Bena, Stefano Giusto, Emil J. Martinec, Rodolfo Russo, Masaki Shigemori, David Turton, Nicholas P. Warner

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Asymptotically-flat supergravity solutions deep inside the black-hole regime

HJE Asymptotically- at supergravity solutions deep inside the black-hole regime Iosif Bena 0 2 4 5 6 7 8 9 Stefano Giusto 0 2 5 6 7 8 9 Emil J. Martinec 0 2 5 6 7 8 9 Rodolfo Russo 0 2 3 5 6 7 8 9 Masaki Shigemori 0 1 2 3 5 6 7 8 9 David Turtong 0 2 5 6 7 8 9 Nicholas P. Warnerh 0 2 5 6 7 8 9 0 Via Marzolo 8 , 35131 Padova , Italy 1 Center for Gravitational Physics, Yukawa Institute for Theoretical Physics 2 CEA, CNRS , F-91191 Gif sur Yvette , France 3 Centre for Research in String Theory, School of Physics and Astronomy 4 Institut de Physique Theorique, Universite Paris Saclay 5 University of Southern California , Los Angeles, CA 90089 , U.S.A 6 High eld , Southampton SO17 1BJ , U.K 7 Kyoto University , Kitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502 Japan 8 Queen Mary University of London , Mile End Road, London, E1 4NS , U.K 9 University of Chicago , 5640 S. Ellis Ave., Chicago, IL 60637-1433 , U.S.A We construct an in nite family of smooth asymptotically- at supergravity solutions that have the same charges and angular momenta as general supersymmetric D1D5-P black holes, but have no horizon. These solutions resemble the corresponding black hole to arbitrary accuracy outside of the horizon: they have asymptotically S3 throats and very-near-horizon AdS2 throats, which however end in a smooth cap rather than an event horizon. The angular momenta of the solutions are general, and in particular can take arbitrarily small values. Upon taking the AdS3 we identify the holographically-dual CFT states. Black Holes in String Theory; AdS-CFT Correspondence - AdS3 First layer of equations: solution-generating technique The solution-generating technique Solution to the rst layer of the BPS equations Second layer of the BPS equations: asymptotically AdS Solution to the second layer of the BPS equations Regularity 4.2.1 4.2.2 Examples 4.3.1 The m = 0 class The structure of the metric Near (r = 0; = 0) Near (r = 0; = =2) 5 Asymptotically- at solutions Novel features of the asymptotically- at solutions 2 3 4 1.1 1.2 1.3 1.4 3.1 3.2 4.1 4.2 4.3 4.4 5.1 5.2 5.3 5.4 6.1 6.2 6.3 The second layer of equations Solving the second layer 5.4.1 5.4.2 Asymptotically- at solutions: regularity and conserved charges 5.5 Conserved charges 1 Introduction and discussion Developing the new class of black-hole microstate geometries Near-horizon geometry The structure of this paper Supersymmetric D1-D5-P solutions to type IIB supergravity CFT states dual to the Asymptotically-AdS solutions The CFT moduli space and the symmetric orbifold Dual states Comparison of conserved charges A Derivation of the explicit form of the function Fk(;pm;q;;ns) B Normalization of CFT states { 1 { 1.1 Introduction and discussion The realization that black holes are thermodynamic black bodies has reshaped our fundamental concept of space and time by introducing profound connections between gravity, quantum mechanics, statistical mechanics and quantum information theory. The need for a dramatic reformulation of our understanding of horizon-scale physics follows from the fundamental con ict between the locality, causality, and unitarity properties of quantum eld theory in the context of black-body (Hawking) radiation emitted by a black hole as described in General Relativity. Over the years, there has been much debate as to which fundamental physical principles need to be relaxed in order to formulate a consistent theory of quantum gravity. Investigation of the entanglement structure of Hawking quanta [ 1, 2 ] has sharpened these issues substantially, showing that one cannot simply use e ective quantum eld theory in the vicinity of a black hole event horizon. Gauge/gravity duality [3] strongly suggests that unitarity must survive as a core principle, at least for the class of examples encompassed by this duality. This is because the space-times on the gravity side of the duality have a time-like boundary and the dual eld theories have a standard unitary quantum-mechanical evolution governed by the Hamiltonian conjugate to the preferred global time coordinate on the boundary. The entire framework of statistical mechanics suggests that the thermodynamic entropy of black holes must be re ected in the statistics of microstate structure. For theories with a gauge/gravity dual, the underlying density of states is that of the quantum Hilbert space. The question then arises as to where and how these microstates are encoded in a black hole. What is the new space-time structure that must emerge at the horizon scale in order to describe a typical black hole microstate? There are many proposals, ranging from fuzzballs [ 1, 4 ], rewalls [2], Bose-Einstein condensates of gravitons [5], webs of wormholes [6] or that the information could be encoded in soft photons around the horizon [7]. The problem is that, with the exception of the fuzzball proposal, none of these proposals has a mechanism that is capable of supporting horizon-scale structure against its rapid and inevitable collapse into the black hole. The fuzzball proposal, and its developments in the microstate geometry programme, replace the horizons of black holes by higher-dimensional, horizonless structures that emerge naturally within string theory. The insistence on horizonless structures comes from requiring that quantum unitarity be preserved [ 1 ]. In terms of the detailed physics, the fuzzball paradigm is that some new phase of matter must emerge at the horizon scale and prevent the formation of the horizon in the rst place. The microstate structure that underpins the black-hole entropy must then remain accessible to outside observers. The fuzzball programme contains a broad range of distinct enterprises and so two of the authors of this paper proposed the following nomenclature [8] to re ne the relevant ideas: 1. A Microstate Geometry is a smooth, horizonless solution of supergravity that is valid within the supergravity approximation to string theory and that has the same mass, charge and angular momentum as a given black hole. { 2 { HJEP02(18)4 2. A Microstate Solution is a formal solution of supergravity equations of motion that is horizonless and that has the same mass, charge and angular momentum as a given black hole. Microstate solutions are allowed to have Planck/string-scale curvatures corresponding to physical brane sources; non-geometric solutions that can be patchwise dualized into a smooth solution are also included. 3. A Fuzzball is the most generic horizonless con guration in string theory that has the same mass, charge and angular momentum as a given black hole. It can involve arbitrary excitations of non-supergravity elds corresponding to massive stringy modes and can be arbitrarily quantum. Microstate geometries, the rst category of microstates above, have been shown to embody the only semi-classical gravitational mechanism known thus far that can support horizon-scale microstructure [9] (see also [10, 11]). From the perspective of holographic eld theory, microstate geometries are intended to capture the infra-red physics of the new phases of matter that emerge at the horizon scale. Thus, one can argue more generally that the e ectiveness of microstate geometries is closely linked to the e ectiveness and accuracy of semi-classical descriptions of holographic eld theory. Building on the work of [4, 12], a growing variety of examples of such geometries have been constructed. These come in two main classes: \bubbled geometries" where all the charges are sourced by Chern-Simons interactions of uxes threading topology [13{17] (see more recently [18{20]); and those in which one of the charges arises from a momentum wave on a bubbled geometry [21{25]. This work culminated in some recent key examples outlined in [26], the details of which we present, and then generalize, in this paper. The examples of microstate geometries constructed to date are still rather limited, and it is not clear whether the most general con gurations are su ciently generic to represent typical microstates of a black hole. They correspond to macroscopic, coherent excitations of a particular set of modes in the supergravity approximation. Furthermore, even if there are a macroscopic number of geometric microstates at extremality, it is not clear whether this property will persist far from extremality, although progress in this direction has recently been made [27{30]. The transition from microstate geometries to the second category | microstate solutions | is expected to encompass more generic horizon-scale microstructure. For instance, in the two-charge system in the D1-D5 duality frame, the microstate geometries involve smooth Kaluza-Klein monopole structures, but the curvature of the typical con guration lies at the scale where the supergravity approximation breaks down [31, 32] (see also [33, 34]). In certain situations adding a third charge has been shown to lower the curvature and smoothen singular two-charge con gurations [24]. Thus it is possible that some portion of the microstate solutions, once fully backreacted, are actually realized as microstate geometries. Microstate solutions also include con gurations that are only patch-wise geometric. See for example [35, 36] for attempts to explicitly construct such microstates, in which di erent patches of spacetime are glued together by U-dualities [37, 38] and which might be related to the backreaction of condensates of stretched branes. { 3 { Finally, the third, \Fuzzball", category is intended to cover the most general situation that can occur in string theory. Examples include condensates of stretched branes [39] that capture a nite fraction of black hole entropy in bubbling microstate geometries [40], and black NS5-branes, whose entropy can be attributed to the Hagedorn phase of \little strings" [41]. However, the proper way to describe the backreaction of condensates of stretched branes is not yet known. String theory contains not only massless supergravity elds but also an in nite tower of massive non-supergravity elds, and it is possible that they are activated in the most general microstates. In particular, massive stringy modes can be excited very near the horizon [42{45], and might distinguish black hole microstates in ways that supergravity cannot. Furthermore, spacetime itself could become highly quantum, so that classical geometric notions such as locality and causality might cease to apply. The divisions between di erent categories are not hard and sharp. For instance, when curvature is of the order of the string scale, there is no clear-cut distinction between supergravity modes and stringy modes. In [46] it was argued that fractionation e ects could lead to a geometry which is stringy as seen by some objects and geometrical as seen by others. Furthermore, one of the authors and Mathur have argued that certain infalling probes interacting with typical fuzzball microstates may for practical purposes experience a smooth horizon, for a subset of physical processes [47, 48]. The important role of microstate geometries in this overall program is that they represent very explicit, computable examples of geometries that are dual to some of the microstates of black holes. Moreover, microstate geometries are capable of supporting extensive microstate structure through classical and semi-classical excitations as well as proving invaluable for the study of more \stringy" microstate excitations, as in [39]. Hence, microstate geometries are the laboratory par excellence for probing and testing ideas about black-hole microstate structure. 1.2 Developing the new class of black-hole microstate geometries One of the problems inherent in the early constructions of microstate geometries was that all known examples carried angular momenta that are large fractions of the maximally allowed value for the corresponding black holes (see for example [16, 49]). This may have led to a misconception that microstate geometries only exist because of a nely-tuned balance between gravity and angular momentum that keeps the constituent branes spread apart. The main mechanism that supports microstate geometries is, in fact, the nontrivial interaction of topological magnetic uxes. This enables such geometries to remain macroscopic and non-singular for arbitrarily small angular momentum. Typical black-hole microstates should also be very well-approximated by the blackhole solution until very close to the horizon. For microstate geometries of extremal black holes this requires a long, BTZ-like, AdS2 throat. To obtain such a throat, prior work used bubbling solutions with multiple Gibbons-Hawking (GH) centers [14, 15]; the moduli space of these solutions includes \scaling" regions [16, 17, 50] in which the GH centers approach each other arbitrarily closely, whereupon the solution develops the requisite long AdS2 throat. Quantum e ects should set an e ective upper bound on the depth of such { 4 { throats [17, 51], and a corresponding lower bound on the energy gap, which matches the lowest energy excitations of the (typical sector of the) dual CFT. All the previously-known scaling microstate geometries involve at least three GH centers [16{20]. Unfortunately, the dual CFT descriptions of these geometries are not yet known. On the other hand, the holographic dictionary between supergravity solutions and CFT states has been constructed for the generic two-charge states [52] and for particular three-charge two-centered solutions [13, 53]. Therefore, we were motivated to construct new three-charge black-hole microstate solutions by adding momentum excitations to a certain two-charge, two-center seed solution. We achieved this using \superstratum" technology [23, 24, 54], which allowed us to introduce momentum-carrying deformations, with speci c angular dependence, that modify the momentum and the angular momenta of the solution without introducing new singularities in the geometry [25, 26]. The geometries in [23, 24, 26] were constructed as excitations of the D1-D5 system in the IIB theory. The holographic duals of the states were identi ed as particular left-moving momentum and angular-momentum modes in the D1-D5 CFT [23, 24, 26, 55]. In [25] these results were generalized to M-theory and the MSW string. The solutions of this paper depend on several parameters. One parameter lowers the angular momenta, while another parameter adds momentum without increasing the angular momenta of the two-charge seed solution. Thus the angular momentum of the solutions can be parametrically small. These deformations therefore allow us to obtain solutions that have arbitrarily small angular momenta and describe microstates of the non-rotating D1D5-P (Strominger-Vafa) black hole. The solutions have an AdS2 throat, which becomes longer and longer as the angular momenta j; ~j ! 0, thus classically approximating the non-rotating black hole to arbitrary precision. 1.3 Near-horizon geometry In many examples of holography in ve and six dimensions, the decoupling limit of the near-horizon geometry is asymptotically a sphere (S2 or S3) bered over AdS3, and gravity is dual to a two-dimensional CFT. In this CFT dual, the asymptotic density of states is governed by the Cardy formula [56], for instance for asymptotically-AdS3 S3 spacetimes, SCFT = 2 r c 6 L0 c 24 j2 + r c 6 ~ L0 c 24 ~j2 : (1.1) This formula holds not only for large charges1 [57]; it remains accurate down to the cosmic censorship bound where SCFT vanishes. At the bound, the naive black hole becomes singular; below the bound, the geometries can have explicit brane sources, or remain smooth and supported by uxes on topological cycles, or can have a combination of both. In this paper we will focus on BPS black holes and so we will only be concerned with the rst term in (1.1). The nave phase diagram is depicted in gure 1 and the parabola at the boundary of the black hole region is the cosmic censorship bound. The Cardy formula indeed shows that increasing the angular momentum takes away from the free 1Large charges mean those satisfying L0 c=24 6j2=c c=6. { 5 { unitarity bound Black Holes N/4 0 SBH= 0 2−charge states N 2JL HJEP02(18)4 above the cosmic censorship bound, i.e. with 6c (L0 jL2, are microstates with the same charges as a black hole with rotation on the S3; states below this bound (depicted in blue) are not. The 1/2-BPS supertube states (on the red line) all lie at or below the bound. energy available to generate black hole entropy, and takes one closer to a solution with a naked singularity. The phase diagram of gure 1 is also an oversimpli cation. Near the cosmic censorship bound, there can be a rich variety of phases involving black holes with other horizon topologies: for instance one can have black holes localized in both AdS3 and the sphere [31, 58]; black holes with supertubes around them; three-charge black rings [59{61]; and multicenter solutions involving black holes, black rings, and supertubes [62, 63]. To avoid the complications of the phase diagram near extremality with macroscopic angular momentum, and get deep into the black-hole regime, one would like to be able to dial the angular momentum to small values, while maintaining a large energy above the ground state, so that the corresponding black hole has a macroscopic horizon area. This was another motivation for constructing the new black-hole microstate solutions outlined in [26]. In six dimensions, the near-horizon geometry of a supersymmetric rotating black string is S3 bered over the extremal BTZ black hole [64], which has the metric ds2BTZ = `2AdS 2 ( dt2 + dy2) + This metric is locally AdS3 and asymptotes to the standard AdS3 form for be written as a circle of radius bered over AdS2 in the near-horizon region for example, [65]). Dimensional reduction on this circle yields the AdS2 of the near-horizon BMPV solution [66]. Following the usual abuse of terminology, we will refer to this region as the AdS2 throat. The BTZ parameters and coordinate are related to the supergravity D1, D5, and P charges Q1, Q5, QP and the radial coordinate r (to be used later) as follows. First, we have (1.2) . It can (see, (1.3) Next, the horizon radius, , of the extremal BTZ solution (1.2) determines the onset of = r 2 `AdS ; `AdS = pQ1Q5 : 2 { 6 { the AdS2 throat (and thus the radius of the bered S1) and is given by charges: the former exerts pressure, thereby expanding the size of the y circle, while the latter exert tension that tries to shrink the circle. There has been a growing interest in the physical properties of microstate geometries [67{70] (see also the recent work [71, 72]). In particular, based on a perturbative analysis, it has been argued that supersymmetric microstate geometries are non-linearly unstable when a small amount of energy is added, potentially leading to formation of a black hole [67], or an approach to typical microstates [69]. We note that the asymptoticallyat solutions of this paper break the isometries that were an intrinsic part of the analysis of [67], so a more detailed analysis is necessary. Furthermore, apparently singular behavior can arise when one oversimpli es the system by ignoring degrees of freedom that are necessary for the correct description of the physics. Therefore the study of these questions requires great care and one must correctly take into account the full phase space of possible con gurations explored by the dynamics. The results of this paper advance our understanding of the phase space of microstate geometries. We intend to investigate questions of stability and their physical interpretation in a future work [73]. 1.4 The structure of this paper A brief summary of some of the new microstate geometries that are asymptotic to AdS3 S appeared in [26]. In this paper we provide a much more detailed description of their construction, and we generalize these solutions to asymptotically- at backgrounds. We work in type IIB string theory on R4;1 S 1 wrapped by n1 D1-branes while n5 D5-branes wrap S1 M, where M is T4 or K3. The S1 is M. We consider the limit where the volume, V4, of M is microscopic and the radius, Ry, of the S1 (parametrized by the coordinate y) is macroscopic, such that the ten-dimensional supergravity brane-charges, Q1 and Q5, are of the same order and macroscopic. In this limit, the D1-branes and D5branes provide a heavy background, in which the momentum P along the y direction is a light excitation. The hierarchy of scales between Ry and V41=4 means that we can reduce the problem to the low-energy, six-dimensional supergravity theory obtained by reduction on M. Following the standard solution-building practice [23], we will consider only the supergravity elds that are both independent of the T 4, or K3, and whose ten-dimensional elds either have no components along M or are proportional to the volume form on M. The result is six-dimensional (1; 0) supergravity coupled to two anti-self-dual tensor multiplets. This system has all the ingredients necessary for the construction of superstrata 3 and has become the workhorse of the microstate geometry programme [23, 25, 26]. Section 2 contains a summary of the six-dimensional supergravity and the equations governing BPS solutions. These equations can be organized in successive layers. A zeroth layer involves non-linear equations de ning the metric of the four-dimensional base space of the solution; in all the solutions in this paper, we will take the same, simple solution for { 7 { this basic layer. The remaining equations are linear and come in two further layers. The rst, which we call Layer 1, is a homogeneous system and, in section 3, we describe how one can nd solutions to this system in two-centered geometries using solution-generating techniques. Since this layer of equations is linear, the most general solution can then be obtained from arbitrary superpositions of the simpler solutions obtained by solutiongenerating methods. The nal layer of BPS equations, which we call Layer 2, is also linear and is sourced by quadratic combinations of the solutions to Layer 1. In section 4 we work in a background that is asymptotic to AdS3 S3 and solve the nal layer of BPS equations when a single mode is excited in Layer 1 of the BPS system. Such single-mode superstrata solutions are structurally much simpler than their multi-mode counterparts [23] but illustrate the major points we wish to make here. In section 5 we then generalize singlemode superstrata to asymptotically- at backgrounds. Readers whose interest lies in the new solutions and their properties, rather than in how they are constructed, may wish to skip directly to sections 4 and 5. Section 6 contains a review of the structure of the CFT that is dual to string theory on AdS3 S 3 M. We identify a particular family of states in the orbifold CFT M N =SN as the dual to our family of microstate geometries; since the states are BPS, this identi cation has meaning even though the states being compared lie in completely di erent loci of the moduli space of the theory. The appendices contain some technical details about the supergravity solutions and the normalization of states in the CFT. The ultimate purpose of this paper is to provide detailed information about the construction of superstrata in both asymptotically-AdS and asymptotically- at space-times. We have provided an extensive introduction so as to set these more technical results in the larger context of the microstate geometry program and we will therefore eschew a conclusions section. 2 Supersymmetric D1-D5-P solutions to type IIB supergravity As we noted in the previous section, we work in type IIB string theory on R4;1 S 1 where M is either T4 or K3. Our solutions are independent of M, and are described by a six-dimensional N = 1 supergravity coupled to two tensor multiplets. The solutions we construct have nontrivial momentum along the common circle wrapped by both the D1 M, and D5 branes, which is parametrized by y and has radius Ry. The rst superstrata [23] were constructed in this theory, which contains all the elds expected from D1-D5-P string emission calculations [74]. The system of BPS equations describing all 1/8-BPS D1-D5-P solutions of this theory was derived in [75]; this is a generalization of the system discussed in [76, 77] and simpli ed in [78]. We work with asymptotically null coordinates u and v, related to y and time t via: u 1 p (t 2 y) ; v 1 2 p (t + y) : (2.1) The BPS solutions have a null isometry along u. The type IIB ansatz comprises the following ingredients. The six-dimensional metric is a bration over a four-dimensional base space B, with metric ds24, which may depend { 8 { on v. The ansatz includes scalars denoted by Z1; Z2; Z4; F ; one-forms on B denoted by ; !; a1; a2; a4; two-forms on B denoted by 1 ; 2; 2; and a three-form on B denoted by x3. All these quantities may depend on v and the coordinates of B. These quantities obey BPS equations that we will display momentarily. We denote the ten-dimensional string-frame metric by ds210, the six-dimensional Einstein-frame metric by ds26, the dilaton by , the NS-NS two-form by B and the RR potentials by Cp. It is convenient to write C6, the 6-form dual to C2, for the purpose of introducing notation. The full ansatz is [75, appendix E.7]: (dv + ) hdu + ! + F (dv + )i + p P ds42 ; 2 Z4 (du + !) ^ (dv + ) + a4 ^ (dv + ) + 2 ; Z2 (du + !) ^ (dv + ) + a1 ^ (dv + ) + 2 ; with 2 ^ (du + !) ^ (dv + ) + x3 ^ (dv + ) ; C6 = vcol4 ^ Z1 (du + !) ^ (dv + ) + a2 ^ (dv + ) + 1 ; Z4 P P Z1Z2 Z1Z2 Z2 4 ; P Z1 Z2 Z42 : (2.2a) (2.2b) (2.2c) (2.2d) (2.2e) (2.2f) (2.2g) (2.2h) (2.3) In the above, ds^24 stands for the at metric on T 4, and vcol4 denotes the corresponding volume form. 2.1 The BPS equations The BPS equations are organized as follows. The four-dimensional metric, ds24, and the one-form satisfy non-linear equations; given a solution to this initial set of equations, the remaining ansatz quantities are organized into two layers of linear equations [75, 78]. In the current paper we build our solutions within a restricted class of solutions to the non-linear layer of equations, in which the four-dimensional base space is R 4 with ds24 the at metric, and in which is v-independent. Given this starting point, the BPS equations for simply impose that it has a self-dual eld strength, d = 4d ; (2.4) where 4 denotes the at R4 Hodge dual. { 9 { To write the remaining BPS equations, let us introduce the 2-forms 1 Da1 + _2 ; Da2 + _1 ; 4 Da4 + _2 : Let us denote the exterior di erential on the spatial base B by d~, and introduce 2 D ~ d 4DZ_1 = D 2 ; 4DZ_2 = D 1 ; 4DZ_4 = D 4 ; D 4 DZ1 = D 4 DZ2 = D 4 DZ4 = 2 ^ d ; 1 ^ d ; 4 ^ d ; In (2.7), the rst equation on each line involves four component equations, while the second equation on each line can be thought of as an integrability condition for the rst equation. The self-duality condition reduces each I to three independent components; in cluding each corresponding equation for ZI makes four independent functional components, upon which there are four constraints. The nal set of BPS equations are linear equations for ! and F , the second of which follows from the vv component of Einstein's equations: D! + 4D! + F d = Z1 1 + Z2 2 2 Z4 4 ; 4D 4 !_ 1 2 DF Z2) 4 (Z_1Z_2 (Z_4)2) 1 Note that F d appears on the left-hand side of the rst equation, so as to separate it from the known sources that arise from the solution to the Layer 1 equations (2.7). 3 First layer of equations: solution-generating technique In this section we describe the construction of the asymptotically-AdS solutions, focusing on Layer 1 of the BPS equations. We will discuss Layer 2 in the next section and the extension of the construction to asymptotically- at solutions in section 5. 3.1 The solution-generating technique Our construction proceeds via the solution-generating technique developed in [23], based on the earlier works [21, 22, 79, 80]. This technique utilizes the symmetry of the simplest two-charge solution: after the change of coordinates corresponding to the CFT spectral ow transformation from the R-R to the NS-NS sector, this solution is nothing but pure AdS3 S3 and thus has an SL(2; R)L SL(2; R)R 2The BPS equations (2.7), (2.8) can also be expressed in a covariant form [23, 25]. 3This symmetry algebra is enhanced to the full Virasoro and current algebras [81], but here we do not consider them and focus on this \rigid" symmetry group. (2.5) (2.6) (2.7) (2.8) One considers a two-charge solution which is a linear (in nitesimal) uctuation around this AdS3 S3 background geometry. We refer to solutions representing such tions as \seed solutions". If one acts on this linear solution with SL(2; R)L generators,4 then, since the background geometry AdS3 S3 is invariant, one generates a new linear uctuation. When written in the original coordinates describing the R-R states, this uctuation has a non-vanishing momentum charge. The original form of the solution-generating technique [79, 80, 83] constructed solutions that involve in nitesimal deformations of AdS3 S3; however we can promote these to solutions involving nite uctuadeformations by using the linear structure of the BPS equations. Concretely, we start with a particular two-charge seed solution that has a non-trivial Z4. The relation of the function Z4 to the pro le function de ning two-charge solutions is reviewed in section 6.2 below; for more details on the pro le function that corresponds to this solution, see [23, eq. (3.10)]. The metric is described in terms of the ansatz quantities described in section 2 as follows. The solution has a at base B = R4 which we write as + (r2 + a2) sin2 d 2 + r2 cos2 d 2 : ds42 = (r2 + a2 cos2 ) dr2 r2 + a2 + d 2 De ning the expression for the one-form is r2 + a2 cos2 ; = p Rya2 2 (sin2 d cos2 d ) : We introduce a real parameter b; to start with, we consider this to be the amplitude of an in nitesimal uctuation, and so we allow ourselves to write a complex phase in Z4 for the moment. The functions and forms of the seed solution at linear order in b are as follows [23, eq. (3.11)]: Z1 = Ry2 a2 Q5 ; Z4 = Ry b ak sink e ik (r2 + a2)k=2 Z2 = Q5 ; ; ! = p Ry a2 2 (sin2 d + cos2 d ) 1 = 2 = 0; 4 = 0; !0 ; where k is a positive integer. To linear order in b, the relation between the parameters a; Ry and the charges Q1; Q5 is 4Acting also with the right-moving part SL(2; R)R SU(2)R breaks supersymmetry, and is not considered in the present paper. Non-extremal linearized solutions where one acts also with SU(2)R have been recently constructed in [82]. a 2 = Q1Q5 Ry2 : (3.1) (3.2) (3.3) (3.4a) (3.4b) (3.4c) (3.5) The \background geometry" obtained by setting b = 0 in the above solution is global AdS3 S3. Indeed, in the new coordinates ~ = t Ry ; ~ = y Ry ; the six-dimensional metric becomes the dual CFT language, the coordinate transformation (3.6) corresponds to the spectral ow transformation from the R-R to the NS-NS sector. We will refer to the coordinate systems (t; y; r; ; ; ) and (t; y; r; ; ~; ~) as the R and NS coordinate systems, respectively. The generators of the SL(2; R)L SU(2)L symmetry of AdS3 S3 are L0 = L 1 = ie Riy (t+y) 2 i 2 r i 2 These satisfy the standard algebra relations Ry 2 r 2 J03 = J 0 = [L0; L 1] = [J03; J0 ] = L 1; J0 ; [L1; L 1] = 2L0; [J0+; J0 ] = 2J03: (3.6) (3.7) The solution-generating technique of [79] adapted to our formulation proceeds as follows: (i) extract the six-dimensional or ten-dimensional elds from the ansatz quantities of the seed solution; (ii) rewrite the elds in the NS coordinate system using (3.6); (iii) act on the elds with the NS generators (3.9) to produce a new linear solution; (iv) use (3.6) again to bring the solution back in the R coordinate system; and nally (v) recast the six-dimensional or ten-dimensional elds into the form of the ansatz, and read o the ansatz quantities. It is cumbersome but straightforward to carry out this procedure starting with our seed solution (3.4). This two-charge solution represents a RR ground state, which can be mapped by spectral ow to an anti-chiral primary state in the NS sector. An antichiral primary is annihilated by J 0 and L1 but generates new (super)descendant states when acted on by J0+ and L 1. So, in step (iii) of the above procedure, we act on the seed solution m times with J + 0 and n times with L 1 ; (3.12) lated by any further action of J0+. where m k, since the action of (J0+)k produces the chiral primary state which is annihi This procedure results in the following ansatz quantities. , Z1;2, 1;2, !, and F , are unchanged at linear order in b from their values given in (3.1), (3.3), (3.4a), (3.4c). Next, Z4 and 4 become: Z4 = b Ry k;m;n e iv^k;m;n ; p 4 = 2 b k;m;n i (m + n) r sin + n where k;m;n a and where (i) (i = 1; 2; 3) are a basis of self-dual 2-forms on R4: (1) (2) (3) + m n k k p 2 v Ry dr ^ d One can check that the elds (3.13) satisfy the Layer 1 BPS equations (2.7). The Layer 2 equations (2.8) are trivially satis ed by ! = !0 and F = 0, because the elds Z4, 4 are in nitesimal and hence the source terms on the right hand side of (2.8) are zero. Let us make a side remark on the CFT state dual to the above solution, to give the reader some rough intuition. The dual holographic description will be fully eshed out in section 6, where the notation used below will be introduced in full. In the NS-NS sector, the above solution corresponds to a component of the CFT state of the form where j00ikNS represents an anti-chiral primary state related to Z4. Spectral- owed to the RR sector, the above component becomes (J0+)m(L 1)nj00ikNS ; (J +1)m(L 1 J 3 1)nj00ikR : In the symmetric orbifold CFT, states generically consist of many strands of di erent lengths. The state j00ikNS;R corresponds to a single strand of length k and the states (3.16) and (3.17) represent their superdescendants. 3.2 The linear solutions for elds (Z4; 4) with quantum numbers (k; m; n) in (3.13), which were obtained by the solution-generating technique, satisfy the Layer 1 BPS equations (2.7). Because these equations are linear di erential equations, we are free to take an arbitrary linear superposition of the solution (3.13), with di erent nite coe cients for di erent values of (k; m; n). Therefore, the following represents a very general class of solutions to the (Z4, 4) rst layer of the BPS equations: Z4 = X k;m;n bk;m;nzk;m;n ; 4 4 = X k;m;n bk;m;n#k;m;n ; 4 where we have de ned the mode functions zk;m;n #k;m;n Ry p (3) cos v^k;m;n : In writing (3.18), we have taken the real part of (3.13). The coe cients b4k;m;n are assumed to be real. More generally we could include di erent phases for di erent values of (k; m; n), but we do not consider that generalization in this paper. The zeroth-layer elds, ds24 and are given by (3.1) and (3.3). In the symmetric orbifold CFT, having a linear combination of di erent modes (k; m; n) corresponds to having multiple strands with di erent quantum numbers (k; m; n) at the same time. Schematically, instead of (3.17), the component of the dual CFT state corresponding to the (Z4, 4) solution (3.18) is now Y h k;m;n (L 1 J 3 1)n(J +1)mj00ikRiNk;m;n ; Nk;m;n / bk;m;n 2 4 : (3.21) The fact that the modes are linear uctuations around AdS3 S3 is re ected in the relation N , which means that this is an in nitesimal excitation above the R ground state. Although the state (3.17) was a superdescendant of the R ground state j00ikR, the state (3.21) is generically not a superdescendant of any R ground state and thus is much more general. We will discuss the form of the CFT states in more detail when we describe the holographic interpretation of these solutions in section 6. Since the Layer 1 equations (2.7) for (Z1; 2) and (Z2; 1) are linear and identical to those for (Z4; 4), we can expand Z1;2, 1;2 in the same modes. Therefore, a very general set of the full Layer 1 elds is given by: bk;m;nzk;m;n ; Z2 = 1 bk;m;nzk;m;n ; Z4 = 2 Z1 = 1 = Q1 + X k;m;n X k;m;n bk;m;n#k;m;n ; 2 X k;m;n X k;m;n bk;m;nzk;m;n ; 4 bk;m;n#k;m;n : 4 (3.22) Q5 + X k;m;n X k;m;n 2 = bk;m;n#k;m;n ; 1 4 = (3.18) (3.19) (3.20) In Z1; Z2, we have included the zero mode parts Q1 , Q5 which correspond to empty AdS3 were obtained assuming that the coe cients bIk;m;n are in nitesimal. However, because the Layer 1 equations are linear di erential equations, even if we make the coe cients I bk;m;n zeroth-layer nite, the elds (3.22) continue to exactly solve the Layer 1 equation when the promote bIk;m;n to be nite parameters and the supergravity con guration (3.22) represents elds, ds24 and , are assumed to be still given by (3.1) and (3.3). So we can a nite deformation of the empty AdS3 S3 background (as far as Layer 1 is concerned). Of course we can perform the same generalization on the CFT side and assume that the numbers of strands, Nk;m;n in (3.21), is of order N . It is then natural to ask whether, for the above nite supergravity deformations, there also exists a simple relation between the CFT and the supergravity parameters. This issue can be clari ed by means of precision holography tests on the 3-point correlators, as discussed in [52, 55, 84]. In particular it is straightforward to generalize this holographic analysis to the new states with n 6= 0 that are the focus of this paper. More concretely, in section 6.2, we will work out the holographic dictionary in detail for some concrete examples and show that the amplitude parameter in Z4 in supergravity, b4k;m;n, is linearly related to the amplitude parameter in CFT; the explicit relation will be given in (6.23). Thus linearity, which is a result of supersymmetry, has allowed us to promote the in nitesimal Layer 1 solution generated in the previous subsection to a nite Layer 1 solution. Once we make bIk;m;n depending quadratically on bIk;m;n. We must compute the Layer 2 quantities, F and !, by solving the Layer 2 di erential equations (2.8) and by requiring that the resulting spacetime is smooth and free of closed timelike curves. These conditions provide constraints on the possible values of the bk;m;n. However it can be quite complicated to make these constraints explicit, since it is usually not obvious how to eliminate singularities in a supergravity solution. In addition, the details of this procedure depend on the choice of the nite, the Layer 2 equations (2.8) require non-trivial solutions Layer 0 elds. A straightforward ansatz for the coe cients bIk;m;n that leads to regular solutions is suggested by the above solution-generating technique, extrapolated to non-linear order [22, 23]. A systematic procedure to construct exact smooth solutions where the scalars ZI have the form (3.22), starts from the two-charge seed in [23, eq. (3.11)], where one keeps also the terms quadratic in b, and acts with a geometry has a this procedure have n = 0, m nite number of non-vanishing modes b4k;m;n. All the modes generated by k, and the b4k;m;n coe cients are not all independent since nite SU(2)L rotation5 by an angle . The resulting they contain only two free parameters b and . One can also observe that this procedure results in b2k;m;n = 0 for any (k; m; n), and hence all the Z2 modes are trivial. However, the modes of Z1 are nontrivial, and depend quadratically on the coe cients b4k;m;n. The relation between the coe cients b4k;m;n and 5Acting with nite SL(2; R)L transformations generates an in nite number of modes and the resulting solution is less easy to analyze. bk;m;n is such that the sources for the second-layer equations (2.8) depend only on the di erence of the modes v^k;m;n v^k0;m0;n0 but not on their sum. The solutions generated in this way are by construction superdescendants of two-charge states and represent only a small subset of the general solutions considered above, where one has modes with arbitrary k; m; n and the coe cients b4k;m;n are arbitrary. One can however exploit the linearity of the rst layer of equations and extrapolate the structure of the coe cients bIk;m;n found for superdescendants to a generic superposition of modes. This is the ansatz that was taken in [23] for constructing superstrata with n = 0, and in the next section we will follow the same approach. 4 Second layer of the BPS equations: asymptotically AdS In this section we describe the construction of solutions to Layer 2 of the BPS equations, focusing on asymptotically-AdS solutions. Asymptotically- at solutions will be presented in the next section. However, before we focus on particular asymptotics, we now make some general remarks outlining some key elements of the structure of the second layer of BPS equations (2.8) that enable us to break the problem into manageable pieces. First, the sources on the right-hand side of (2.8) are quadratic in the Z's and 's, which means that the sources involve the sums and di erences of their Fourier mode dependences, v^k;m;n. Explicitly, there are two types of source: those with phase dependence v^k+k0;m+m0;n+n0 , and those with phase dependence v^k k0;m m0;n n0 (here we assume that k k0 0 without loss of generality). As mentioned at the end of the previous section, for superdescendant states one nds no sources with phase v^k+k0;m+m0;n+n0 . Furthermore, based on experience [23], when mode dependences v^k;m;n add together, the corresponding solution to Layer 2 (2.8) is generically singular. In this paper we will always arrange that these Layer 2 sources are absent. Thus our strategy will be to set the b2-modes to be zero, and to tune the b1-coe cients so as to cancel the terms with v^k+k0;m+m0;n+n0 in the Layer 2 sources. Note that for a pair of modes (k; m; n) and (k0; m0; n0), such a cancellation is not possible if (km0 k0m)(kn0 k0n) 6= 0, unless one excites other elds. Thus, if one allows generic modes to interact, the construction of regular solutions could prove rather more challenging. By adjusting the Fourier coe cients in (Z1; 2) in terms of those in (Z4; 4) in this way, one can construct fully smooth microstate geometries. This tuning of Fourier coe cients to create a smooth outcome is known as \coi uring" [85{87]. Since the sources of Layer 2 are quadratic in ZI and I , the b1 coe cients depend quadratically on the b4 coe cients. We emphasise that the Fourier coe cients b4k;m;n of Z4 are allowed to remain arbitrary, in agreement with the results of IIB string scattering amplitudes [74, 88{90]. We will see that this choice makes the source terms in the Layer 2 equations particularly simple, and leads to smooth solutions. Because of the obstruction mentioned above, this approach is not directly applicable to the most general superposition of modes, depending on both m and n. However, interactions between multiple Fourier modes were considered in [23] for n = 0 and that approach should work whenever each pair of modes satis es (km0 k0m)(kn0 k0n) = 0. In particular, we expect that the construction of multi-mode solutions with m = m0 = 0 will be possible using methods very similar to those employed in [23]. To keep things simple, in this paper we will only construct solutions with a single mode, for which this issue does not arise. For a single Fourier mode, there will be terms with phase dependence v^2k;2m;2n, and there will be \RMS" modes, proportional to the square of the Fourier coe cient (b4k;m;n)2 but independent of (v; ; ). We will deal with each separately. The non-oscillating RMS terms depend only upon (r; ) and the contributions to ! and F from these terms simplify to: !RMS = !1(r; ) d + !2(r; ) d ; F RMS = F (r; ) : (4.1) HJEP02(18)4 (4.2) As we will see, these equations can be solved completely, albeit in a form involving sums of multinomial coe cients. Physically, these RMS parts of the solution contain the longerdistance e ects of the oscillations, encoding all the resulting changes (with respect to the seed solution) in the asymptotic momentum charge and angular momenta. To solve the equations for oscillating sources one can use a gauge invariance of (2.8) to set6 F Having made this gauge choice, one can write (2.8) in terms of di erential operators acting on each component of !. From experience [23], one typically nds that this system can rst be reduced to a Laplacian on the sum of components (! + ! ), and once this equation is solved, with a little guesswork one can leverage this to nd the complete solution for all the components of !. We will describe this procedure in more detail in section 5.3. With only a single mode, and for asymptotically-AdS solutions, the coi uring results in a complete cancellation of the mode dependence in the metric. Hence the metric is completely independent of (v; ; ). In these solutions, the tensor elds still oscillate as functions of (v; ; ), but the coi uring cancels these oscillations in the energy-momentum tensor and so the gravitational eld does not oscillate. The gravitational eld does respond to the uctuations, but only through their RMS e ects. Thus, the single-mode asymptotically-AdS superstrata which we construct in this section are the simplest of their kind, and their second-layer equations (2.8) have only non-oscillating, RMS sources. To obtain asymptotically- at superstrata, one must \add 1's" to Z1 and Z2, and this creates new source terms that depend explicitly upon the oscillations in (v; ; ). This requires us to nd new families of solutions to (2.8). These solutions will be constructed in section 5 and, as we will see, their metric will depend non-trivially upon (v; ; ) even after coi uring. 4.1 Solution to the second layer of the BPS equations Following [23] we set the oscillations in (Z2; 1) to zero, since this choice emerges naturally from the non-linear solution-generating method described at the end of section 3.2. We also 6Note that this choice is only possible for modes that have a non-trivial v-dependence, of the type we will consider in this paper. specialize to a single-mode superstratum, which means reducing to single Fourier modes in (3.22). The structure of the quadratic sources in Layer 2 means that it is natural for the modes of (Z1; 2) to have twice the mode numbers of (Z4; 4). Since we now specialize to a single mode, we suppress the (k; m; n) indices on bIk;m;n. Thus we take the full Layer 1 elds to have the form: with Z1 = Q1 + Z4 = Ry b4 1 = 0 ; b1 Ry2 2Q5 With these choices, the sources of the Layer 2 BPS equations have an oscillating part that depends only upon v^2k;2m;2n as well as an RMS part. As in [23], we nd that such oscillating sources generically lead to singular angular momentum vectors, !. However, the Fourier coe cient of the oscillating source is proportional to b1 b24 and so we take: This coi uring of the modes removes the singular oscillating parts and leaves us with only the RMS sources. As we will see, this leads to a smooth solution. The solution for ! and F is now given by the sums of the original supertube solutions and the solution for the RMS pieces, as in (3.4c) and (4.1): !AdS = !0 + !RMS ; F = F RMS : The equations (2.8) for !RMS now reduce to: d!RMS + 4d!RMS + F d = p 2 Ry b42 2k;2m;2n m(k + n) (2) n(k k m) (3) ; Lb F = where Lb is the scalar Laplacian on the base space B: LbF 1 r + the components !r = ! = 0. We write Inspired by the results of [23, 91], we de ne !RMS k;m;n(d + d ) + k;m;n(d d ) : Since the right-hand side of (4.7) has no component in the (1) direction, we can set 1 sin cos k n(k k m) 2 # 2k;2m+2;2n 2 ; ^k;m;n k;m;n + p Ry r2 + a2 sin2 Fk;m;n + Ry b42 p (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) (4.9) (4.10) (4.11) HJEP02(18)4 + r2 cos 2 a2s2 r2 + a2s2 p 2Ry r (r2 + a2s2) b 24 ms2 + nc2 r(r2 + a2) sin 2 r2 + a2s2 + p Ry sin 2 2 (r2 + a2s2) b 2 4 k;m;n r sin 2 r2 + a2s2 k cos 2 a2r2(r2 + a2) cos 2 2k;2m;2n Fk;m;n ; k;m;n mn k Fk;m;n : mr2 + n(r2 + a2) (2r2 + a2) 2k;2m;2n where Fk;m;n F is the solution of (4.8). Then ^k;m;n satis es Lb ^k;m;n = Rpy b42 1 1 (k m)2(k + n)2 2 (r2 + a2) cos2 k2 (nm)2 k2 2k;2m+2;2n + k;m;n has been computed, k;m;n is determined by substituting (4.10) into (4.7), which gives (s = sin , c = cos ) the equation F2k;2m;2n = where (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) To solve the equations for F and ^k;m;n, we must nd the function F2k;2m;2n that solves LbF2k;2m;2n = 2k;2m;2n In appendix A, we nd that the solution to this problem is given by In terms of F2k;2m;2n, the form of F Fk;m;n and k;m;n for general k; m; n is Fk;m;n = 4b42 m2(k + n)2 k2 k;m;n = Rpy b42 (k 2 m)2(k + n)2 k2 n2(k m)2 k2 F2k;2m;2n + F2k;2m+2;2n 2 ; F2k;2m+2;2n + F2k;2m;2n 2 m2n2 k2 r2 + a2 sin2 4 b4 2Fk;m;n 4 xk;m;n : 4 L3 (2k;2m;2n) = sin L4 (2k;2m;2n) = L0 (2k;2m;2n)F b and 2 r sin cos r cos r cos (k + n) (m + n)(r2 + a2) !^r + sin (m + n)r2 n !^ p 4 2 (m + n) Ry div(2k;2m;2n)!^ ; L0 (2k;2m;2n)F b + (r2 + a2) sin2 (k m)2 sin2 + m2 cos2 Fb ; (k + n) (m + n)(r2 + a2) !^1 Given that the BPS solution is u independent, any BPS solution is invariant under the u ! u + U (xi; v) ; ! ! ! dU + U_ ; F ! F 2 U_ ; (k p F b ! Fb + 2 2 (m + n) Ry f ; !osc + f m)(r2 + a2) a2(k + n) sin2 d Using the sources (5.3) and (5.4) we arrive at the following equations: (m+n) r sin + n 1 m k r sin ; p 2 Ry a2 r sin 2 Fb ; (5.8) HJEP02(18)4 (5.10) (5.11) (5.12) L1 L2 L3 L4 (2k;2m;2n) = (2k;2m;2n) = (2k;2m;2n) = (2k;2m;2n) = p p p 2 Ry 2k;2m;2n following reparametrization of u: This leads to the gauge invariance: + r2 cos2 (m + n)r2 n !^2 : 2 b 2 4 b 2 4 b 2 4 b1 b1 b1 b 2 4 b1 Q5 b1 Q5 b1 Q5 b1 n b1 Q5 m + 1 ; 1 ; m k : n k div(2k;2m;2n)!osc (5.9) mr2 na2 cos2 d cos v^2k;2m;2n ; for any function, f (r; ). For our oscillating modes we will use this gauge invariance to set: Fb = F The standard route to solving the system (5.10) is to observe that the equations involving (2k;2m;2n) do not involve derivatives of !^r and !^ . One then uses these equations to obtain expressions for !^r and !^ and then substitutes them back into the other two equations to obtain two second order di erential equations for !^1 and !^2. Rather remarkably, one then nds that the combination !^1+!^2 satis es a straightforward harmonic It is elementary to solve this and we nd the following particular solution: Ry p b1) b1 k2 Q5 The next step is slightly more of an art than a science. The individual equations for !^1 and !^2 separately are very complicated. However, based on experience, the form of !^1 + !^2, and how !^1 and !^2 should behave in various limits, one is naturally led to !^1 = !^2 = Ry p p b1) b1) a2 r 2 + b1 n(k Q5 Q5 k2 b1 m(k + n) m) k2 : ; These manifestly add to (5.15), however these expressions are not the solutions for general (k; m; n) but they are solutions for either m = 0 or m = k. Thus we will preserve the appearance of m in our formulae with the understanding, for the moment, that we are considering m = 0 or m = k. Presumably there are more complicated recurrence relations for solutions with intermediate values of m. Armed with expressions for !^1 and !^2, one can now substitute back into the equations in (5.10) involving L2 (2k;2m;2n) and L3 (2k;2m;2n) and solve for !^r and !^ algebraically. The general result is a mess, but there are simple formulae that work for m = 0; k: !^r = p So far (5.16), (5.17) and (5.13) de ne complete solutions for !osc for m = 0 and m = k. (5.16) (5.17) b1 a2 (m + n) Q5 b1 a2 (m + n) Q5 k k = 0 : : The careful reader might note that we have added a seemingly redundant m(k m) term to the expression for !^ . This is because if one substitutes (5.16), (5.17) and (5.13) into (5.10) then the result either vanishes or is proportional to Thus (5.16), (5.17) and (5.13) provides a solution for all (k; m; n) provided that As we will see below, this is the new coi uring constraint required by regularity of the solution and so we actually have the complete, regular solution for all (k; m; n) ! The way we rst arrived at this complete solution was to nd the coi uring constraint for m = 0 and m = k, and from this we inferred the general coi uring relation (5.19). Then we used !^1 and !^2 in (5.10) to solve for !^r and !^ algebraically and then imposed (5.19). This led to the complete expressions for !^r and !^ . The complete solution for !osc is given by (5.6) with components given by (5.16) and (5.17). Putting the components together and using the coi uring constraint (5.19), we can simplify !osc to: !osc = b1 Ry Q5 2 2 p (m + n) a2 cos2 m(k + n) Finally, we note that the coi uring condition may be re-written as: b1 1 + a2 m + n Q5 k = b42 : + + + 2k;2m;2n k r 2 (m + n) k n(k m) k2 n(k m) k2 cos v^2k;2m;2n d + a2 n(k k2 cot k2 m) m(k + n) k2 cos v^2k;2m;2n d 1 r(r2 + a2) sin v^2k;2m;2n dr tan sin v^2k;2m;2n d ) : (5.18) (5.19) (5.20) (5.21) (5.22) This form is useful because a2=Q5 is a small dimensionless parameter in the neardecoupling limit. given by: 5.4 Asymptotically- at solutions: regularity and conserved charges The complete asymptotically- at solution to the second layer of the BPS equations is ! = !0 + !RMS + !osc ; F = F RMS ; where the individual pieces are given by (3.4c), (4.10) and (5.20). The general conditions for regularity have been discussed in section 4.2. We verify here that these conditions are satis ed also by the asymptotically- at extension of our solutions. We focus on the two potentially problematic points: the center of R4 (r = 0; = 0) and the supertube location (r = 0; = =2). 5.4.1 = 0) In the asymptotically- at solution the one-form ! acquires a new contribution !osc that depends on v^2k;2m;2n, and the analysis of its behaviour around the point (r = 0; = 0), where the polar coordinates degenerate, requires some extra care. Notice rst of all that !osc is nite at (r = 0; = 0), since the 1=r and 1= sin poles inside the curly bracket in (5.20) are canceled by 2k;2m;2n. This is not enough however to conclude that !osc is smooth: we should check that its components with respect to a local orthonormal frame are nite. Switching to the coordinates (r~; ~) and v~ de ned in (4.19) and (4.20), we nd n(k m) k2 1 k2 Im e i(m+n) 2p2 v~ Ry dr~ r~ m)(r~ sin ~ei )2(k m) d(r~ cos ~ei )2n + cot 2~d~ + cos v^2k;2m;2n (d + d ) +n(r~ cos ~ei )2n d(r~ sin ~ei )2(k m) (5.23) (5.24) (5.25) Since r~ sin ~ei and r~ cos ~ei are linear combinations of well-behaved Cartesian coordinates around (r = 0; = 0), the identity above shows that !osc is smooth at the center of space. Near r = 0; = =2, one can make the coordinate transformation where we consider to be a small parameter. Recalling that near r = 0; = =2, 2k;2m;2n behaves like r = a cos ; 2 sin 2k;2m;2n r2n(cos )2m ; and noting that we always have at least one of n or m greater than zero, we see that in the above coi ured !osc there are no terms that scale as 1 when terms start at 0. Thus, near = 0, ! is well-approximated by !RMS. ! 0 and that the rst Therefore the requirement that the 1= terms in the metric near = 0 vanish is the same as in the asymptotically-AdS solutions, and leads to the constraint (4.23). Having ensured this, the solution is smooth in the neighborhood of = 0. 5.5 Conserved charges The global charges are read o from the asymptotically- at solution in a straightforward way. The oscillating terms average to zero when integrated over the S1 and hence give vanishing contributions to the global charges. Only the RMS modes, which were derived in section 4, are therefore relevant for this computation. Moreover, since the interaction between di erent modes produces terms with a non-trivial v-dependence which also do not contribute to the charges, the relations valid for general multi-mode solutions are given by simply summing the contributions of the single modes that we write below. the one-form + !: One nds of which we know a closed form expression for any k; m; n, given in (3.3), (3.4c), (4.18). J = Ry 2 1 1# J~ = Ry a2 : 2 p J 2 n The D1 and D5 supergravity charges Q1 and Q5 are given by the 1=r2 terms in the large r expansion of the warp factors Z1 and Z2. As was noted before, regularity imposes the constraint (4.23) on Q1 and Q5. The dimensionful momentum charge Qp is likewise encoded in the function F as F 2Qp=r2. The expansion of (4.17) gives 2k k m 1 k + n 1 n 1 : The dimensionful angular momenta J , J~ can be extracted from the + component of One can check that the charges computed from the asymptotically- at solution are identical to those obtained from the AdS geometry. These can be compared with the charges of the dual CFT states. In section 6, we will see that the supergravity and CFT charges agree if we assume simple linear relations between the amplitude parameters in supergravity, a; b4k;m;n, and the corresponding parameters in CFT. The most signi cant feature of our solutions is that they can be taken to lie deep within the black hole regime n1n5np j2 > 0, i.e. the regime of parameter space where black holes with a regular horizon exist. We observe that our solutions lie within this bound for b 2 a2 > k n + p(k m + n)(m + n) : 6 CFT states dual to the Asymptotically-AdS solutions The geometries we have constructed have macroscopic brane charges. As is usual in gauge/gravity duality, one can go to a region of the moduli space where the geometry near the branes decouples from the ambient spacetime, and correspondingly the dynamics on the branes decouples from gravity in the asymptotically- at region. Quantum gravity in the near-source geometry is then dual to a non-gravitational theory [3]. The asymptoticallyAdS3 solutions of section 4 are dual to states in the 2d CFT that arises as the low-energy limit of the gauge theory on the underlying system of branes. In the next subsection we review some basic properties of this CFT and in sections 6.2 and 6.3 we identify the CFT states dual to the geometries we construct. 6.1 The CFT moduli space and the symmetric orbifold In the weak-coupling limit of the dynamics of n5 D5-branes, the n1 D1-branes bind to the D5 branes by dissolving into them as instanton strings [3, 92, 93]. The corresponding CFT (5.26) (5.27) (5.28) (5.29) is thus often thought of as a sigma model on the moduli space of n1 instantons in U(n5) gauge theory on M = T4 or K3. This description of the CFT is however an approximation adapted to a particular corner of the CFT moduli space. Consider for instance supergravity compacti ed on T 4 Sy1; it has a moduli space E6(6) USp(8) . E6(6)(Z) : The decoupling limit takes Ry=`str ! 1 holding the energy scale ERy and the T4 volume xed; one is e ectively going to the cusp in the moduli space where Ry is asymptotically large, and in particular pQ1Q5 branes, the limit isolates the region r2 Q1; Q5. moduli in the cusp parametrize the space The decoupling limit breaks the duality symmetry to SO(5; 5; Z), and the remaining SO(5; 5) SO(5) SO(5) . SO(5; 5; Z) : Ry. In the geometry sourced by the (6.1) (6.2) (6.3) (6.4) The 27 of wrapped brane and momentum charges on T4 S1 splits up into 10 16 1, where the 10 consists of branes wrapping S1 which become in nitely heavy in the decoupling limit, and thus are part of the background data of the CFT; the 16 consists of the assortment of branes wrapping T4 but not S1; and the 1 is the momentum charge on S1. The background of n5 D5-branes and n1 D1-branes breaks the duality symmetry further; of the 25 moduli in (6.2), ve are frozen by the attractor mechanism [ 94, 95 ], and the duality group is broken to the subgroup H of the duality \little group" SO(5; 4; Z) which xes the ten-component M = T4 the CFT has a 20-dimensional moduli space of couplings background charge vector . Similar considerations hold for M = K3. In the end, for MX = SO(5; 4) SO(5) SO(4) . H : The structure is conveniently seen by isolating an SO(2; 2; Z) = SL(2; Z)L SL(2; Z)R subgroup of the modular group that acts on the moduli = C0 +i=gs by gR fractional linear transformations and ~ = C4 + iv4=gs by gL fractional linear transformations (when all the other antisymmetric tensor moduli are set to zero). The background charges (n~1; n1; n~5; n5) of fundamental and D-strings, NS5 and D5-branes, respectively, can be packaged into a matrix Q = n~1 n1! n5 n~5 which transforms under duality as Q ! gLQgRt and in particular preserves N = det(Q); we are interested in the duality frames where n~1 = n~5 = 0 and n1n5 = N . The attractor mechanism then relates and ~ via ~ = d1=d5. orbifold locus (3,2) (2,3) HJEP02(18)4 = C0 + i=gs (here mapped from the upper half-plane to the Poincare disk) has a cusp for every decomposition of N into two factors (n1; n5) such that N = n1n5. This slice of the moduli space is the fundamental domain of the congruence subgroup 0(N ) of SL(2; Z); n5 copies of the SL(2; Z) fundamental domain meet at the cusp corresponding to backgrounds with n5 vebranes. Here we illustrate the structure for N = 6. The moduli space has a cusp for every factorization of the integer N into a pair of integers n1 and n5 [96, 97], see gure 3.8 One sees this from the duality rotation with an5 bn1! 1 1 gL = gR = a b ! n1 n5 which maps the D1-D5 charges (n1; n5) to (n1n5; 1), and relates a cusp at = nb5 to the cusp at = 0. Note that these two cusps are always separated by na1 at = i1, and a cusp at nb5 = n11n5 . While this is not a duality transformation that preserves the background, the fact that the moduli space is a symmetric space under the action of continuous duality rotations in SO(5; 4) means that if there is a cusp for a particular choice of charges (n1; n5), then there is another cusp with the charges (n1n5; 1), or for that matter any pair of integers whose product is N . To get from one to the other involves moving a macroscopic distance through the moduli space from one cusp to another. In each cusp, there is a codimension-four singular locus where the system is neutrally stable and can fragment by breaking apart into separate charge centers [96, 97]. For instance, the long string sector of perturbative string theory in AdS3 S3 [96, 98], which describes fundamental strings propagating out to the AdS3 boundary in the background of electric and magnetic NS 3-form ux, is precisely such an instability. This pathology can (6.5) = na1 to the cusp n1 and n5 are coprime, so that the brane background is truly bound and cannot fragment into smaller pieces. be avoided by turning on any of the four moduli (for instance the combination of C0 and C4 which preserves the xed scalar condition) that take the theory away from the singular locus. In the slice of the moduli space depicted in gure 3, there is a singular locus at <( ) = 0 (the red dashed line) and at each of the images of this line under the maps that permute the weak-coupling cusps (in this example, the magenta dashed arc between = 1=2 and = 1=3). The description of the CFT in terms of a sigma model on the moduli space of n1 instantons in U(n5) gauge theory is an approximate weak-coupling description in a particular cusp, corresponding to a particular choice of factorization. In the cusp where n5 = 1 and n1 = N , there is a (codimension four) weak-coupling locus where the sigma model target space X is the symmetric product orbifold [99, 100] (see also the review [101]) X0 = M N =SN (6.6) which is a solvable conformal eld theory. Note that the map (6.5) does not imply that there is a symmetric orbifold description for every cusp; in fact it is rather unlikely that there is one. The analysis in [97] of the masses of states carrying conserved charges in the 16 of branes and momenta on the T4 showed that the energetics was consistent with the corresponding charges in the symmetric orbifold only if the latter was a weak-coupling limit in the cusp where n1 = N and n5 = 1. The sigma model on the moduli space of instantons may be a weak-coupling description of other cusps, but it does not reduce to the symmetric product orbifold at low energies. The symmetric orbifold is a nonsingular, parity-invariant CFT. In the cusp correalong the orbifold locus. sponding to n5 = 1 the parity-invariant points are at C0 = 0 and C0 = 1=2. The former is the singular locus, which leaves C0 = 1=2 as the orbifold locus. The SL(2; Z) map (6.5) from the cusp at = i1, corresponding to the symmetric orbifold, to a cusp at macroscopic charges (n1; n5) (where the supergravity description is valid) has an5 bn1 = 1. The cusp is a macroscopic distance in the natural hyperbolic metric (j=d j)22 from any point na1 , with The regions of the moduli space admitting a low-energy supergravity description are distant from the solvable locus X0, and hence it is not possible in general to relate states in the solvable CFT with particular supergravity backgrounds. Nevertheless for BPS states one can compare quantities such as conformal dimensions and three-point correlators, which are protected by supersymmetry against renormalization as we move across the moduli space [102]. In this section we provide a dictionary between the asymptotically-AdS geometries of section 4 and particular CFT states in the RR sector of the orbifold CFT. This dictionary should be interpreted in the following sense: the three-point correlators between these RR states and any chiral primary operator can be calculated either holographically using the supergravity solutions, or at the orbifold point using the free- eld realization of the CFT, and the two results match. This point of view was introduced in [52] in the sector of the RR ground states that are dual to two-charge geometries, and was extended in [55] to the three-charge geometries of [23]. Of course for non-protected quantities, such as four-point functions, the e ects of wavefunction renormalization generically become visible and the relation between the gravity solutions and the orbifold CFT states described here becomes less useful. With this understood, we now identify and discuss the holographic dictionary; our notation and conventions mostly follow [24, 55]. The twisted-sector ground states of the symmetric orbifold (M)N =SN CFT in the RR sector are 14 -BPS, and map to known supergravity supertube geometries [103, 104]. There is an independent twisted sector for each conjugacy class in the symmetric group. Symmetric group elements consist of words which are products of (non-overlapping) cyclic permutations of the copies of M. The conjugacy class of a word is characterized by the number Nk of cycles of length k in the word, with the total length (including cycles of length one) being P k kNk = N . When k copies of the CFT on M are sewn together by a cyclic permutation boundary condition, the result can be thought of as the CFT on M on the k-fold covering of the coordinate cylinder on which the CFT lives. The supersymmetric ground states of the k-cyclic twisted sector are thus the same as those of M. For M = T4, these ground states consist of ultrashort multiplets labelled by spin-1/2 doublets , _ under the SU(2) SU(2) R-symmetry, and A; B under an auxiliary SU(2)A: j _ ik ; jABik ; j Bik ; jA _ ik ; The highest-weight states of the rst two of these multiplets are bosonic, while in the last two they are fermionic. We will focus on two ground states in particular | the highestweight state j++ik of the R-symmetry bispinor multiplet (the rst one in (6.7)), and the singlet combination of the auxiliary SU(2)A bispinor (the second one in (6.7)), j00ik ABjABik : k Y k;s fNksg jsik Nks : Z2 = 1 + Z1 = 1 + Q5 Z L L L 0 jxi 0 L p 2 A + B 0 jxi 1 gi(v0)j2 dv0 ; jxi gi(v0)j2 gi(v0)j2 dv0 ; ! = A p 2 B Q5 Z L jg_i(v0)j2 + jg_5(v0)j2 dv0 ; Q5 Z L g_j (v0) dxj The full ground state is then a tensor product of ground states for the cyclic twists in the symmetric group conjugacy class, having N (s) copies of k-cycle ground states (6.7) of the polarization state s. We often refer to the cycles of the symmetric product as `strands' of the dual CFT. The class of states we are interested in thus takes the form The role of the various polarizations of cyclic twist is illustrated by the map between the 14 -BPS states and their dual geometries [52, 55, 84]: (6.7) (6.8) (6.9) (6.10a) (6.10b) (6.10c) Z4 = a1 = a4 = x3 = 0 ; (6.10d) where the dot on the pro le functions indicates a derivative with respect to v0, L 2 Q5=Ry, and 4 is the Hodge dual with respect to the at R 4 metric ds24 = dxidxi. One can expand the two-charge pro le functions in Fourier series g1 + ig2 = g3 + ig4 = X `>0 X `>0 a " a`++ e 2Li` v0 + a` e 2Li` v0 ; e 2Li` v0 + ` e 2Li` v0 ; a + g5 = Im `>0 ` X a`00 e 2Li` v0 : subject to the constraint on the overall amplitude X ` ja`++j2 + ja` j2 + ja`+ j2 + ja` +j2 + ja`00j2 = Q1Q5 : Ry2 The speci c solutions of section 4 are built starting from the ground states 1 a ; a00 k bk = b4k;0;0 with all other coe cients equal to zero. As we see from (6.10), the numbers Nki of cycles with polarization i _ j _ ik in the number eigenstates (6.9) determine the amplitudes of the Fourier coe cients of the functions gi(v) and thus specify gyrations of the brane bound state in the four dimensions transverse to its worldvolume. Having only j++i1 strands corresponds to a round supertube rotating in the x1-x2 plane. The j00ik strands carry no transverse angular momentum, and so do not a ect the shape of the supertube. Their numbers N 00 do however determine the amplitudes of the Fourier coe cients of the function g5 which speci es the harmonic function Z4 and therefore a ects the antisymmetric tensor elds of the supergravity background. Because the elds of the supergravity solution have both a well-de ned amplitude and k phase, they are represented as coherent states built from the number eigenstates (see for instance equations (3.6){(3.12) of [55]). of the 14 -BPS ground state [23]. The three-charge states dual to the geometries of sections 3 and 4 are built on these unexcited (m = n = 0) round supertubes. The momentum-generating excitation labelled by m in supergravity adds JL charge and P charge in equal proportion to the harmonic function Z4; one can identify it as corresponding to the action of J +1 on the j00i strands Under spectral ow to the NS-NS sector, j00ik is mapped into an anti-chiral primary state j00ikNS with h = j3 = k=2, and J +1 is mapped to J0+. Because j00ikNS is the lowestweight state of SU(2)L, it can be acted on by J0+ a maximum of k times, which means that k. Similarly, the generalization to n > 0 involves additional CFT excitations which carry n units of momentum but no angular momentum; it is natural to identify them with the mode operator (L 1 J 3 1), which commutes with J +1. This discussion is completely (6.11) HJEP02(18)4 (6.12) (6.13) fNk(s)g N1 Y k;m;n X k;m;n 4 X0 in parallel to the one we gave on the gravity side in section 3. Thus we are led to the set of states j++i1 (J +1)m (L 1 m! n! j00ik (6.14) must satisfy9 This is the more precise version of the \intuitive" formula that we presented in (3.21). The numbers fN1; Nk;m;ng specify the number of strands with particular quantum numbers and N1 + kNk;m;n = N : (6.15) principle include all the other ground states in (6.7).10 In (6.14), we considered only the ground state j00ik with excitations on it, but we can in The classical supergravity dual does not correspond to the state (6.14) with xed numbers fN1; Nk;m;ng but rather to its coherent superposition [52, 84, 104]. We introduce a set of dimensionless parameters fA1; Bk;m;ng, which are closely related to the supergravity mode amplitudes a and bk;m;n of (3.18). The state dual to the coi ured supergravity solution can be written, generalizing the n = 0 expression in [55], as (fA1; Bk;m;ng) = AN1 1 Y (Bk;m;n)Nk;m;n k;m;n # fN1;Nk;m;ng ; (6.17) where the sum is restricted to fN1; Nk;m;ng satisfying (6.15). In the large N limit this sum is dominated by a stationary point fN 1; N k;m;ng which can be found by calculating the norm j (fA; Bk;m;ng)j2 and taking its variation with respect to fN1; Nk;m;ng. In order to do this, we need to derive the e ect of the momentum-carrying perturbations J +1 and (L 1 J 3 1) on the normalization of the state (6.14). For n = 0 the result is given in equation (3.17) of [55] and the generalization to n 6= 0 is given in appendix B. Using the result, the saddle-point values are found to be m n + k 1 jBk;m;nj2 : (6.18) Thus far, we have been considering the general set of states that have strands with different quantum numbers (k; m; n); namely, Nk;m;n 6= 0 for multiple sets of values (k; m; n). 9On the supergravity side, this constraint can be understood as the level-matching constraint on the worldsheet of the F1-P supertube which is in the same duality orbit as the 14 -BPS D1-D5 supertube ground state. 10The generalization of (6.14) and (6.15) to include all ground states is fNks;m;ng Y k;m;n;s m! (J+1)m (L 1 J3 1)n jsik n! Nks;m;n ; X k;m;n;s kNks;m;n = N : (6.16) From this perspective, the numbers N1 and Nk;m;n in (6.14) should more consistently be denoted by N1+;0+;0 and Nk0;0m;n, respectively. The supergravity solutions dual to the more general states will have base space data, (B; ), that is more complicated than the base space used in this paper. In [24], another set of special states for which the data (B; ) remain simple (called \Style 1" states) are discussed. Now, let us focus on the special states (6.14) where Nk;m;n is non-zero only for one particular set of values (k; m; n), which can be written as N1;Nk;m;n j++i1 N1 (J +1)m (L 1 m! n! j00ik In this expression, k; m; n are not summed over, but are xed numbers. The corresponding coherent state (6.17) can be written in terms of two quantities A1; Bk;m;n as (A1; Bk;m;n) = X0 N1;Nk;m;n A1N1 (Bk;m;n)Nk;m;n N1;Nk;m;n ; where the two numbers N1; Nk;m;n satisfy N1 + kNk;m;n = N : (6.19) (6.20) (6.21) (6.22) (6.24) (6.25) (6.26) We propose that the states (6.20) are the holographic duals of the single-mode supergravity superstrata that we constructed in section 4. The saddle point values for A1; Bk;m;n are determined by (6.18). If we substitute N1; Nk;m;n in (6.21) with their saddle point values, we obtain jA1j2 + m n + k 1 jBk;m;nj2 = N: If we compare this with (4.23), we nd that the dimensionless coe cients A1, Bk;m;n of the CFT are related to the corresponding Fourier coe cients a and b4k;m;n in supergravity via jA1j = R s N Q1Q5 a ; jBk;m;nj = R s N 2Q1Q5 m bk;m;n : (6.23) The explicit proposal for the CFT states dual to the microstate geometries we constructed allows one to perform quantitative AdS/CFT studies that generalize those of [55]. We leave such an interesting investigation for future work. 6.3 Comparison of conserved charges We can now compare the CFT parameters to those of the supergravity solutions. From the expression for Z1 in (6.10) we see that the D1 charge of the 14 -BPS ground states is The supergravity charges Q1, Q5 are related to the quantized D1 and D5 numbers, n1 and where V4 is the coordinate volume of T 4. The relation between Qp and the quantized momentum number np is given by n5, by Qp = (2 )4 np gs2 04 V4Ry2 Q1Q5 Ry2N np : Q1 = Q5 Z L L 0 Q1 = (2 )4 n1 gs 0 3 V4 jg_i(v0)j2 + jg_5(v0)j2 dv0: Q5 = n5 gs 0 ; The dimensionful angular momenta J , J~ de ned in (5.27) are related to the quantized ones j, ~j by J = (2 )4gs2 04 V4 Ry j = Q1Q5 j ; RyN J~ = (2 )4gs2 04 ~j = V4 Ry Q1Q5 ~j : RyN (6.27) By using the dictionary between bulk and CFT quantities introduced in the previous section it is possible to match the supergravity and CFT calculations of the conserved charges and of the three-point functions of chiral primary operators [55, 103, 104]. Here we focus on the conserved charges; these can be derived by using the average number of each type of strands derived in (6.18). For instance, in the class of states we considered, each strand of the type j00ik carries (m + n) units of momentum, thus the total momentum is equal to (m + n) times the average number, N k;m;n np = (m + n)N k;m;n = Ry2N " Q1Q5 m + n 2k m (6.28) In the last step we used (6.18) and (6.23) in order to show that the result matches perfectly (5.26). Similarly for the angular momenta, we nd n Ry2N 2Q1Q5 k m j = ~j = 1 2 1 2 N 1 + mN k;m;n = N 1 = Ry2N 2Q1Q5 a2 ; which exactly match the supergravity results in (5.28). Acknowledgments We thank Samir Mathur, Harvey Reall and Jorge Santos for discussions. The work of IB and DT was supported in part by John Templeton Foundation Grant 48222 and by the ANR grant Black-dS-String. The work of SG was supported in part by the Padua University Project CPDA144437. The work of EJM was supported in part by DOE grant DESC0009924, and a FACCTS collaboration grant. The work of RR and MS was supported in part by the Science and Technology Facilities Council (STFC) Consolidated Grants ST/P000754/1 and ST/L000415/1. This work of MS was supported in part by JSPS KAKENHI Grant Numbers 16H03979, and MEXT KAKENHI Grant Numbers 17H06357 and 17H06359. The work of DT was supported by a CEA Enhanced Eurotalents Fellowship and a Royal Society Tata University Research Fellowship. The work of NPW was supported by DOE grant DE-SC0011687. For hospitality during the course of this work, EJM, MS and NPW are grateful to the IPhT, CEA-Saclay; and EJM, DT and IB are grateful to the Centro de Ciencias de Benasque. SG, EJM, RR, MS, DT, and NPW thank the Yukawa Institute of Theoretical Physics, Kyoto University for hospitality during the workshop \Recent Developments in Microstructures of Black Holes" (YITP-T-17-05) where this paper was completed. Derivation of the explicit form of the function Fk(;pm;q;;ns) In constructing the solution to the Layer 2 equations, one encounters the problem of nding the function Fk(;pm;q;;ns)(r; ) satisfying Lb (p;q;s)Fk(;pm;q;;ns) = k;m;n ; where k;m;n and the scalar Laplacian with wave numbers (p; q; s), Lb in (3.14) and (5.9) respectively. In this appendix we derive the explicit form of the solution Fk(;pm;q;;ns)(r; ). In section 4.1, we gave the explicit expression for F2k;2m;2n F2(k0;;20m;0);2n. The derivation below is a straightforward generalization of the derivation of Fk(;pm;q) done in ref. [23]. For some intermediate steps that are not spelled out in the derivation below, see (p;q;s), are de ned Let us rst de ne Gk;m;n = k;m;n ; Sk;m;n = k;m;n : It is straightforward to check that these functions satisfy the following recursion relation: Lb (p;q;s)Gk;m;n = (n2 s2)Sk+2;m+2;n 2 + ((p + s)2 (k + n + 2)2)Sk+2;m+2;n q)2)Sk;m+2;n + (m2 q2)Sk;m;n : Introducing the generating functions (A.1) (A.2) (A.3) (A.4) (A.5) we can rewrite the equation we want to solve, (A.1), as F ( ; ; ) G( ; ; ) S( ; ; ) X k;m;n X k;m;n X k;m;n Fk(;pm;q;;ns) ek +m +n ; Gk;m;n ek +m +n ; Sk;m;n ek +m +n ; Lb (p;q;s)F ( ; ; ) = S( ; ; ) ; and the recursion relation (A.3) as h Lb (p;q;s)G( ; ; ) = e 2 2 +2 ((@ + 2)2 s2) + e 2 2 ((p + s)2 (p q2)iS( ; ; ) : (A.6) F = h (p + s)2) (p s ) 2 i 1 G 1 " e (p + s)2 (p + s)2 (p + s)2 G : (A.7) Expanding in a multinomial expansion and examining the coe cient of ek +m +n , j3 2) { zn+(n+}| 1) { zn (n }| 1) X 4 i=0 j1+j2+j3=i j1 j1; j2; j3 2) { z (m { (zk m )(k }| m 1 } | j2 1)(}m| j1 (k+ + n+)(k+ + n+ 1) (k + n )(k + n 1) Gk 2(j1+j2+1);m 2(j2+1);n 2j3 1 X 1)! (k+ + n+ (k+ + n+)! Gk 2(j1+j2+1);m 2(j2+1);n 2j3 i j1) (k (m 1)! j3)! (n 1)! (k + n n ! j2 m (k + n )! 1) j3 1)! j1; j2; : : : ; jn jn! 2 p m m 2 where is the multinomial coe cient, and where we de ned de nition (A.9), we nd that the explicit expression for Fk(;pm;q;;ns)(r; ) is In fact, the sum can be simpli ed because Pi1=0 Pj11+j2+j3=i = P1 j1;j2;j3=0. Using the (A.8) (A.9) (A.10) (A.11) Fk(;pm;q;;ns) = 1 4(k+ + n+)(k + n ) X j1;j2;j3=0 k++n+ j1 j2 j3 1 j1 j2 j3 1 j1; m j2 1; n j3 k++n+ 1 k+ m+; m+ 1; n+ k Gk 2(j1+j2+1);m 2(j2+1);n 2j3 q i = j1 + + jn n n 2 s j1; j2; j3 k +n m k +n m ; m 1 1; n where the sum is over 0; min(k+ + n+; k + n ) 1: (A.12) In particular, when p = q = s = 0, F2(k0;;20m;0);2n = 1 4(k + n)2 j1+j2+j3 k+n 1 X j1;j2;j3=0 k m j1; m j2 1; n j3 k+n 1 k m; m 1; n 2 2 B Normalization of CFT states In this appendix, we compute the normalization of the CFT states (6.14). Because the states we consider are obtained by exciting the 1/4-BPS states (6.9), it is useful to recall the norm of (6.9): NST j fNksgj2 = N ! Qk;s Nks! kNks : This given by the number of ways one can partition N to obtain the desired distribution of strands; for details, see section 3 (in particular eq. (3.4)) of [55]. The normalizations of the excited states obtained by the action of J +1 and (L 1 J 3 1) are determined in terms of those of the ground state, (B.1), through the commutation relations of the N = 4 superconformal algebra, [Lm; Ln] = (m n)Lm+n + [J ma; Jnb ] = i abcJ mc+n + m m; n ab ; nJ ma+n ; 2 2 J = J 1 iJ 2 and consider the following state: where k is the level of the SU(2) current algebra and c = 6k is the Virasoro central charge. On a strand of length k, the positive integer k is indeed the level of the diagonal sum of the k copies of the N = 4 algebra being wound together by the Zk cyclic twist. De ne J1 (J +1)mj00ik = 2J03 + k + J +1J1 (J +1)m 1j00ik ; the J03 operator evaluates to m 1 acting on the right. Proceeding iteratively one arrives at J1 (J +1)mj00ik = 2 X ` + mk (J +1)m 1j00ik m 1 `=0 1) (J +1)m 1j00ik : Iterating this again for (J1 )m acting from the left, one nds kh00j(J1 )m (J +1)mj00ik = m! k 1) k (m 2) k = m! (k (B.1) (B.2) (B.3) (B.4) (B.5) One nds similarly J13) (L 1 J 3 1)nj00ik 2L0 2(n 2 2J03 + + (L 1 J 3 1)(L1 J13) (L 1 1) + k + 2(n 2) + k + + 2(0) + k = n k + (n 1) (L 1 J 3 1)n 1j00ik ; J 3 1)n 1j00ik i (L 1 J 3 1)n 1j00ik once again iterating for the nth power of the lowering operator one nds J 3 1)nj00ik = n! k + (n 2) k + (0) = n! (k + n (k 1)! 1)! : Combining the results (B.1), (B.5), (B.7), one nds the norm of the state (6.14): j fN1;Nk;m;ngj2 = N ! N1! k;m;n Y 1 1 Nk;m;n! k m n + k 1 n : The classical supergravity dual does not correspond to the state (6.14) but rather to its coherent superposition (fA1; Bk;m;ng) given in (6.17). The norm of this state is, using the results above, j (fA1; Bk;m;ng)j2 = X0 jA1j2N1 h Y jBk;m;nj2Nk;m;n i k;m;n N ! N1! k;m;n Y 1 1 Nk;m;n! k m n + k 1 n where the sum is over fN1; Nk;m;ng satisfying the constraint (6.15). In the large N limit, the sum is dominated by a stationary point fN 1; N k;m;ng, which can be obtained by setting to zero the variation with respect to fN1; Nk;m;ng of the summand and using the Stirling formula. The result is N 1 = jAj2 ; m n + k 1 n jBk;m;nj2 ; which is a generalization of equation (3.21) of [55]. The strand multiplicities fN1; Nk;m;ng are not independent variables but satisfy the constraint (6.15). However this constraint applies to the average values fN 1; N k;m;ng and so we have jAj2 + X k;m;n m n + k 1 n jBk;m;nj2 = N : Open Access. 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Iosif Bena, Stefano Giusto, Emil J. Martinec, Rodolfo Russo, Masaki Shigemori, David Turton, Nicholas P. Warner. Asymptotically-flat supergravity solutions deep inside the black-hole regime, Journal of High Energy Physics, 2018, 14, DOI: 10.1007/JHEP02(2018)014