Habemus superstratum! A constructive proof of the existence of superstrata

Journal of High Energy Physics, May 2015

We construct the first example of a superstratum: a class of smooth horizonless supergravity solutions that are parameterized by arbitrary continuous functions of (at least) two variables and have the same charges as the supersymmetric D1-D5-P black hole. We work in Type IIB string theory on T 4 or K3 and our solutions involve a subset of fields that can be described by a six-dimensional supergravity with two tensor multiplets. The solutions can thus be constructed using a linear structure, and we give an explicit recipe to start from a superposition of modes specified by an arbitrary function of two variables and impose regularity to obtain the full horizonless solutions in closed form. We also give the precise CFT description of these solutions and show that they are not dual to descendants of chiral primaries. They are thus much more general than all the known solutions whose CFT dual is precisely understood. Hence our construction represents a substantial step toward the ultimate goal of constructing the fully generic superstratum that can account for a finite fraction of the entropy of the three-charge black hole in the regime of parameters where the classical black hole solution exists.

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Habemus superstratum! A constructive proof of the existence of superstrata

Received: March Habemus superstratum! A constructive proof of the existence of superstrata Iosif Bena 1 2 4 7 8 9 Stefano Giusto 1 2 4 6 8 9 Rodolfo Russo 1 2 3 4 8 9 Masaki Shigemori 0 1 2 4 5 8 9 Nicholas P. Warnerg 1 2 4 8 9 CEA Saclay 1 2 4 8 9 F- 1 2 4 8 9 Gif sur Yvette 1 2 4 8 9 France 1 2 4 8 9 Open Access 1 2 4 8 9 c The Authors. 1 2 4 8 9 0 Hakubi Center, Kyoto University 1 Queen Mary University of London , Mile End Road, London, E1 4NS, U.K 2 Via Marzolo 8 , 35131 Padova , Italy 3 Centre for Research in String Theory, School of Physics and Astronomy 4 Universit`a di Padova , Via Marzolo 8, 35131 Padova , Italy 5 Yukawa Institute for Theoretical Physics, Kyoto University 6 INFN - Sezione di Padova 7 Institut de Physique Th eorique 8 Los Angeles , CA 90089 , U.S.A 9 Yoshida-Ushinomiya-cho , Sakyo-ku, Kyoto 606-8501 , Japan We construct the first example of a superstratum: a class of smooth horizonless supergravity solutions that are parameterized by arbitrary continuous functions of (at least) two variables and have the same charges as the supersymmetric D1-D5-P black hole. We work in Type IIB string theory on T 4 or K3 and our solutions involve a subset of fields that can be described by a six-dimensional supergravity with two tensor multiplets. The solutions can thus be constructed using a linear structure, and we give an explicit recipe to start from a superposition of modes specified by an arbitrary function of two variables and impose regularity to obtain the full horizonless solutions in closed form. We also give the precise CFT description of these solutions and show that they are not dual to descendants of chiral primaries. They are thus much more general than all the known solutions whose CFT dual is precisely understood. Hence our construction represents a substantial step toward the ultimate goal of constructing the fully generic superstratum that can account for a finite fraction of the entropy of the three-charge black hole in the regime of parameters where the classical black hole solution exists. existence; of; superstrata; Black Holes in String Theory; AdS-CFT Correspondence; Supergravity Mod- 1 Introduction Supergravity background The IIB solution The M-theory and five-dimensional pictures The equations governing the supersymmetric solutions Outline of the construction of a superstratum Solving the first layer of BPS equations The solution generating technique A rigidly-generated three-charge solution A general class of solutions to the first layer A three-charge ansatz The first type of source The second type of source Example 1: (k1, m1) = (k2, m2) Example 2: (k2, m2) = (1, 0); (k1, m1) arbitrary Example 3: k1 = m1 + 1, m2 = 1 The second layer Regularity, asymptotically-flat superstrata and their charges The CFT description Basic features of the dual CFT 14 -BPS states and their descendants A class of superstrata: the CFT description Discussion, conclusions and outlook A D1-D5 geometries B Solution of the (generalized) Poisson equation There has been growing evidence that string theory contains smooth, horizonless boundstate or solitonic objects that have the same charges and supersymmetries as large BPS black holes and that depend on arbitrary continuous functions of two variables. These objects, dubbed superstrata, were first conjectured to exist in [1], by realizing that some of the exotic brane bound states studied in [2]1 can give rise to non-singular solutions in the duality frame where the charges of these objects correspond to momentum, D1-branes It was subsequently argued that, assuming that superstrata existed, the most general class of such objects could carry an entropy that scales with the charges in exactly the same way as the entropy of the D1-D5-P black hole, and possibly even with the same coefficient [5]. Since this entropy would come entirely from smooth horizonless solutions, this would substantiate the fuzzball description of supersymmetric black holes in string theory: the classical solution describing these black holes stops giving a correct description of the physics at the scale of the horizon, where a new description in terms of fluctuating superstrata geometries takes over. Partial evidence for the existence of superstrata can be obtained by analyzing string emission in the D1-D5 system [6, 7], or by constructing certain smaller classes of supergravity solutions [813]. However, to prove that superstrata indeed exist, one needs to explicitly construct smooth horizonless solutions that have the same charges as the D1-D5-P black hole and are parameterized by arbitrary continuous functions of two variables, which is a The purpose of this paper is to construct such solutions and thus demonstrate that superstrata exist. Furthermore, we will be able to find precisely the CFT states dual to these solutions and show that these states are not descendants of chiral primaries, which means that they are much more general than all the known solutions whose CFT dual is precisely understood [8, 10, 14, 15]. This is a huge step toward achieving the ultimate goal of constructing all smooth horizonless solutions that have the right properties for reproducing the black-hole entropy and thus proving the fuzzball conjecture for BPS black holes. Our procedure relies on the proposal [1] that superstrata can be obtained by adding momentum modes on two-charge D1-D5 supertubes: supertube solutions [1618] have eight supercharges and are parameterized by functions of one variable; adding another arbitrary function-worth of momentum modes to each supertube was argued to break the supersymmetry to four supercharges and result in a superstratum parameterized by arbitrary continuous functions of two variables. However, as anybody familiar with supertube solutions might easily guess, trying to follow this route brings one rather quickly into a A simpler route to prove that superstrata exist is to start from a maximally-rotating supertube solution and try to deform this solution by making the underlying fields and 1In [2], double supertube transitions [3] of branes were argued to lead to configurations that are parametrized by functions of two variables and are generically non-geometric. For further developments on exotic branes see [4]. metric wiggle in two directions. This approach is attractive for several reasons. First, the holographic dictionary for the 14 -BPS (8-supercharge)2 D1-D5 supertubes is well understood [19, 20] and so, as we will describe later in this paper, we can then generalize this dictionary to the 18 -BPS (4-supercharge) D1-D5-P superstrata. Second, the equations that govern the superstrata solutions are well-known [21, 22], and can be organized in a linear fashion [23], and so this technique appears to be the technique of choice, all the more so because it has enabled the construction of solutions that depend of two arbitrary functions each of which depends upon a different variable [12]. Nevertheless, while extensive trial and error has led to many solutions that depend on functions of two variables they have all, so far, been singular.3 The key ingredient simplifying the task of smoothing the singularities of these solutions is a fourth type of electric field that appears neither in the original five-dimensional U(1)3 ungauged supergravity, where most of the known black hole microstate solutions have been built [2426], nor in the six-dimensional uplift in [21, 23], where the solutions of [12] were constructed. The presence of this field can drastically simplify the sources that appear on the right-hand sides of the equations governing the superstratum and allows us to find smooth solutions depending on functions of two variables in closed form. The solutions with this field can only be embedded in a five-dimensional ungauged supergravity with four or more U(1) factors, or in a six-dimensional supergravity with two or more tensor multiplets. Fortunately, the equations underlying the most general supersymmetric solution of the latter theory were found in [27] and these equations can also be solved following a linear algorithm similar to the one found in [23]. The essential role for this fourth type of electric field in the solutions dual to the typical microstates of the D1-D5-P black hole was first revealed by analyzing string emission from the D1-D5-P system [6, 7] and from D1-D5 precision holography [19, 20]. Furthermore, in [28, 29] it was shown that adding this field to certain fluctuating supergravity solutions can make their singularities much milder.4 The fact that the extra field plays an important part in both obtaining smooth, fluctuating three-charge geometries and in the description of D1-D5-P string emission processes is, in our opinion, no coincidence, but rather an indication that the solutions we construct are necessary ingredients in the description of the typical microstates of the three-charge black hole. Our plan is to start from a round supertube solution with the fourth electric field turned on and to prove that this solution is part of a family of solutions that is parameterized by functions of two variables. There are two natural perspectives on these solutions. The first is to recall that, in the D1-D5 duality frame, the infra-red geometry of the two-charge supertube solution is AdS3global S3. This background has three U(1) symme2Throughout this paper N1 -BPS will denote a state with 3N2 supercharges. 3It is important to remember that our purpose is to reproduce the black hole entropy by counting smooth horizonless supergravity solutions, or at most singular limits thereof, that one can honestly claim to describe in a controllable way. If we were to count black hole microstate solutions with singularities, we could easily overcount the entropy of many a black hole. 4This has allowed, for example, the construction of an infinite-dimensional family of black ring solutions that gives the largest known violation of black-hole uniqueness in any theory with gravity [29]. Gibbons-Hawking R 3 base. The Lunin-Mathur two-charge supertube solutions [16, 18], as well as their generalizations that have the fourth type of electric field turned on [8, 19, 20], correspond to shape deformations of the supertube, and their shapes and charge densities these classes of solutions are parameterized by functions of one variable and, as such, correspond to special choices of spherical harmonics on the three-sphere of the round supertube solution. Our superstrata will depend non-trivially upon all three angular coordinates, but only through a two-dimensional lattice of mode numbers (defined in (3.23)). The second perspective comes from decomposing the functions of two variables that parametrize our superstratum solutions under the SU(2)L SU(2)R isometry of the S3 and the SL(2, R)L SL(2, R)R isometry of the AdS3. The shape modes of the twocharge supertube preserve eight supercharges and have SU(2)L SU(2)R quantum numbers spin of the field in the theory, each Fourier mode is determined essentially by one quantum number. Thus, these solutions are parameterized by functions of one variable, as expected. The solutions we construct have four supercharges and correspond to adding leftmoving momentum modes to the supertube. The generic mode will have SL(2, R)L weight h > 0. Since h is independent of j, these will generate intrinsically two-dimensional shape modes on the S3. Since the equations underlying our solutions can be solved using a linear algorithm, superposing multiple spherical harmonics gives rise to very complicated source terms in the equations we are trying to solve. Furthermore, most of the solutions one finds by brute force give rise to singularities. In the earlier construction of microstate geometries, such singularities were canceled by adding homogeneous solutions to the equations. Here we will see that this technique does not allow us to obtain smooth solutions from a generic superposition of harmonics on the S3 in all electric fields, and that we have to relate the combinations of spherical harmonics appearing in the electric fields. At the end of the day, the resulting smooth solutions will contain one general combination of spherical harmonics on a three-sphere, which can be repackaged into an arbitrary continuous function Our superstratum can be precisely identified with a state at the free orbifold point of the D1-D5 CFT. The dual CFT interpretation, besides providing a crucial guide for the supergravity construction, firmly establishes that our solutions contribute to the entropy of the three-charge black hole, and clarifies what subset of the microstate ensemble is captured by our solutions. In the previous literature, all three-charge geometries with a known CFT dual [8, 10, 14, 15] had been obtained by acting on a two-charge solution (in the decoupling limit) with a coordinate transformation that does not vanish at the AdS3 boundary. On the CFT side this is equivalent to acting with an element of the chiral algebra on a Ramond5Here we are considering the Ramond-Ramond (RR) sector. Ramond (RR) ground state, and produces a state which is identified with a descendant of a chiral primary state in the Neveu-Schwarz-Neveu-Schwarz (NSNS) sector. In contrast, the microstate solutions we construct here cannot be related, generically, to two-charge microstate solutions via a global chiral algebra rotation. They thus do not correspond to descendants of chiral primaries but represent much more generic states than the ones previously considered in [8, 10, 14, 15]. In the interests of full disclosure, while the results presented here represent a major step forward in the microstate geometry programme, it is also very important to indicate what we have not yet achieved. First, the superstratum solutions we construct in this paper are still rather coarsely grained in that they do not fully capture states in the twisted sector of the dual CFT (see section 7). That is, while we do indeed have a superstratum that fluctuates nontrivially as a function of two variables, the fluctuations we construct here are dual to restricted classes of integer-moded current-algebra excitations in the dual CFT and so, at present, our superstrata solutions do not have sufficiently many states to capture the black-hole entropy. Thus, we have not yet achieved the holy grail of the microstate One should also note that typical states will contain general combinations of fractionalmoded excitations in a twisted sector of very high twisting, corresponding to a long effective string of length equal to the product of the numbers of D1 and D5 branes. This sector of the CFT might not be well described within supergravity. However, to prove the validity of the microstate geometry programme it is sufficient to show the existence of a superstratum which contains general fractional modes in twisted sectors of arbitrary finite order; this will establish the existence of a mechanism which allows to encode the information of generic states in the geometry. The fact that, in the limit of very large twisting, corrections beyond supergravity might have to be taken into account does not invalidate the existence of such a mechanism. In particular, we hope that in subsequent work we will be able to refine the mode analysis and the holographic dictionary obtained in this paper and obtain superstrata containing general fractional modes. The other, more technical issue is that the systematic procedure given in this paper does not yet provide a complete description of the solution for all combinations of Fourier modes of the arbitrary function of two variables that parametrizes the superstratum. As yet, we have not been able to obtain the closed expression for one function that appears in some components of the angular momentum vector. In principle these could be singular, but we do not expect this, for two reasons: first, we have the general explicit solution for one of the components of the angular momentum vector and this component is regular and, from our experience, if there are singularities in the angular momentum vector they always appear in this particular component. Secondly, we have actually been able to find this function and construct the complete solution for several (infinite) families of collections of Fourier modes. These families were chosen so as to expose possible singular behaviors and none were found. Thus, while we do not have explicit formulae for one function that appears in the angular momentum vector for all combinations of Fourier modes we believe that this is merely a technical limitation rather than a physical impediment. The construction presented in this paper establishes that the superstratum exists as a bound state object of string theory, and that its supergravity back-reaction gives rise to smooth horizonless three-charge solutions. Having shown this, we believe that a fully generic superstratum is within reach and thus one will be able to show that a finite fraction of the entropy of the BPS black hole comes from smooth horizonless solutions. This, in turn, would imply that the typical states of this black hole will always have a finite component extended along the direction of the Hilbert space parameterized by horizonless solutions, and hence will not have a horizon. Thus one would confirm the expectations and goals of the fuzzball/firewall arguments:6 the horizon of an extremal supersymmetric black hole is not an essential, fundamental component but the result of coarse-graining multiple horizonless configurations. More broadly, we would like to emphasize that results presented here provide a remarkable confirmation of the power of the approach we have been using to establish that there is structure that replaces the horizon of a black hole: we have directly constructed this structure in supergravity. As we emphasized in [46], this approach could have failed at many different stages throughout its development. The most recent hurdle has been to show that supergravity has structures that might contain enough states to count the entropy of the black hole. In [5] we have argued that this can happen if string theory contains three-charge superstrata solutions that can be parameterized by arbitrary continuous functions of two variables. The present paper shows explicitly that these solutions exist and furthermore that they are smooth in the duality frame where the black hole has D1,D5 and momentum charges. (It was the successful clearing of this latest hurdle that led to our somewhat celebratory title for this paper.) Though most of the recent literature on the information paradox has focused on Alice-and-Bob Gedankenexperiments, we believe that general quantum information arguments about physics at a black-hole horizon will always fall short of resolving the paradox: failure is inevitable without a mechanism to support structure at the horizon scale. It is remarkable that string theory can provide a natural and beautiful solution to this essential issue and, as was shown in [47], microstate geometries provide the only possible gravitational mechanism and so must be an essential part of the solution to the paradox. In section 2 we introduce the six-dimensional supergravity theory where our D1-D5-P microstate solutions are constructed and also recall the connection of these solutions to those constructed in the more familiar M2-M2-M2 duality frame. We write the equations governing the supersymmetric solutions of the six-dimensional supergravity theory in a form that highlights their linear structure and simplify the problem by choosing a flat four-dimensional base space metric. The equations governing the supersymmetric solutions can then be organized in a first layer of linear equations, which determine the electric and magnetic parts of the gauge fields associated with the D1- and D5-branes, and a second layer of linear but inhomogeneous equations, which determine the momentum and the angular momentum vectors. In section 3 we solve the first layer of equations. We start from a round D1-D5 supertube carrying density fluctuations of the fourth type of electric field and apply a CFT 6See [3045] for some developments in that area. symmetry transformation to generate a two-parameter family of modes that carry the third (momentum) charge. We then use the linearity of the equations to build solutions that contain arbitrary linear combinations of such modes. Section 4 contains the most challenging technical part of the superstratum construction: finding the solution of the second layer of equations. We explain how the sources appearing in these equations have to be fine tuned to avoid singularities of the metric, and how this requirement selects a restricted set of solutions to the first layer of equations. These solutions are parameterized by certain coefficients that can be interpreted as the Fourier coefficients of a function of two variables, which defines the superstratum. We then construct the general solution for the particular component of the angular momentum 1-form that, from our experience, controls the existence of closed timelike curves. We also find in section 5 the remaining components of this 1-form, thus deriving the complete solution for several (infinite) families of collections of Fourier modes. We verify the regularity of the solutions in these examples. Although we mostly work in the decoupling regime, in which geometries are asymptotic to AdS3 S3, in section 6 we present a way to extend our solutions and obtain asymptotically five-dimensional ( R4,1 S1) superstrata geometries. We also derive the asymptotic charges and angular momenta of these geometries. These results are then used in section 7 to motivate the identification of the states dual to the superstrata at the free orbifold point of the D1-D5 CFT. We point out that states dual to our superstrata are descendants of non-chiral primaries and we show how some of the features of the gravity solution have a natural explanation in the dual CFT. Section 8 summarizes the relevance of our construction for the black-hole microstate geometry programme and highlights possible future developments. Several technical results are collected in the appendices. In appendix A we recall the form of general two-charge microstates and in appendix B we explain how to use a recursion relation to solve some of the differential equations of the second layer. Readers who are not so interested in the gory technical details of our solutions can simply read sections 2 and 3 in order to understand the supergravity structure that we use in constructing the explicit superstratum solution, and read section 7 in order to understand the corresponding states in the dual CFT. Supergravity background The existence of the superstratum was originally conjectured based upon an analysis of supersymmetric bound states within string theory. The ( 12 -BPS) exotic branes of string theory were thoroughly analyzed in [2, 4], where it was also argued that objects carrying dipole charges corresponding to such branes can result from simple or double supertube transitions. In [1] it was pointed out that the hallmark of these bound state objects is that they are locally 12 -BPS, but when they bend to form a supertube they break some of the supersymmetry. In particular the objects that result from a simple supertube transition are 14 -BPS and are parameterized by arbitrary functions of one variable, while the objects that result from a double supertube transition are 18 -BPS and are parameterized by arbitrary functions of two variables. As explained in [2, 4], most of the double supertube transitions result in objects carrying exotic brane charges, which are therefore non-geometric. However, in [1] it was pointed out that when D1 branes, D5 branes and momentum undergo a double supertube transitions the resulting 18 -BPS object is not only geometric but also potentially giving rise to a class of smooth microstate geometries parameterized by arbitrary functions of two variables. This object became known as the superstratum. Thus, this fundamental bound state in string theory could, as a microstate geometry, provide a very large semiclassical contribution to the 18 -BPS black-hole entropy. Indeed it was argued in [5] that a fully generic superstratum could capture the entropy to at least the same parametric growth with charges as that of the three-charge black hole. Thus the construction of a completely generic superstratum has become a central goal of the microstate geometry programme. The supertube transitions that yield the superstratum were analyzed in detail in [1] and it was shown that indeed such solitons could be given shape modes as a function of two variables while remaining 18 -BPS. Based on the forms of these supertube transitions it was argued that the resulting geometry should be smooth but this remained to be substantiated through computation of the fully-back-reacted geometries in supergravity. Since this initial conjecture, much progress has been made in finding the supergravity description of the superstratum. The structure of the BPS equations led to the construction of doubly fluctuating, but singular BPS, superthreads and supersheets in [9, 11]. Simple but very restricted classes of superstrata were obtained in [12]. In parallel with this, string amplitudes were used to very considerable effect to find the key perturbative components of the superstratum [6 The fact that the BPS equations underlying the superstratum are largely linear [23] means that knowledge of the perturbative pieces can be sufficient for generating the complete solution. Finally, in an apparently unrelated investigation of new classes of microstate geometries [28] and new families of black-ring solutions [29], a mechanism arising out of the perturbative superstrata programme was used to resolve singularities and find new physical solutions. We are now in a position to pull all these threads together and obtain, for the first time, a non-trivial, fully-back-reacted smooth supergravity superstratum that fluctuates as a function of two variables. We begin by reviewing the basic supergravity equations that need to be solve, starting in the D1-D5-P duality frame and discussing how this reduces to an analysis within six-dimensional supergravity. While we will be working with the T 4 compactification of IIB supergravity to six dimensions, it is important to note that in our supergravity theory in six dimensions without vector multiplets. This implies that all our supergravity results may be trivially ported to IIB supergravity on K3. The IIB solution form [27, appendix E.7]: The general solution of type IIB supergravity compactified on T 4 same supercharges as the D1-D5-P system and is invariant under rotations of T 4 has the ds62 = B = C0 = C2 = C4 = Z4 vcol4 Z4 (du + ) (dv + ) + a4 (dv + ) + 2 , Z2 (du + ) (dv + ) + a1 (dv + ) + 2 , C6 = vcol4 Z1 (du + ) (dv + ) + a2 (dv + ) + 1 P Z1 Z2 Z42 . Here ds210 is the ten-dimensional string-frame metric, ds26 the six-dimensional Einsteinuseful to also list C6, the 6-form dual to C2, to introduce all the quantities entering the supergravity equations.) The flat metric on T 4 is denoted by ds24 and the corresponding volume form by vcol4. The metric ds24 is a generically non-trivial, v-dependent Euclidean metric in the four non-compact directions of the spatial base, B. We have traded the usual t y u = v = two-forms on B; and x3 is a three-form on B . All these functions and forms can depend not only on the coordinates of B but also on v. As discussed below, if the solution is v-independent, the one-forms a1, a2, a4 may be viewed as five-dimensional Maxwell fields. Finally, C is a v-dependent top form in B which can always be set to zero by using an appropriate gauge [27]. To preserve the required supersymmetry, these fields must satisfy BPS equations [27] and thus get interrelated to one another as we will explain in subsection 2.3. Note that we use the fact that the internal manifold of our solutions is T 4 only as an intermediate technical tool, but the final solutions we obtain are solutions of six-dimensional supergravity with two tensor multiplets, which can describe equally well microstate geometries for the D1-D5-P system on K3. The M-theory and five-dimensional pictures Three-charge microstate geometries are expected to be smooth only in the D1-D5-P duality frame, in which we exclusively work in this paper. However, it is useful to make connection to other duality frames that are probably more familiar to the reader, in particular the M-theory frame in which all the electric charges are on the same footing and described by M2-branes. Moreover, by compactifying M-theory on T 6 and truncating the spectrum one supergravity coupled to n vector multiplets. However, it is important to note that the M-theory and D1-D5-P frames are different in one crucial respect: v-dependent solutions in the D1-D5-P frame, which are essential ingredients of the superstratum conjecture, are not describable in the M-theory frame, because the T-duality along the common D1-D5 direction, which connects the two frames, transforms v-dependent solutions into solutions that contain higher KK harmonics and therefore cannot be described by supergravity. Therefore, for the purposes of the current paper, the M-theory picture explained here should be regarded as a book-keeping device to understand the degrees of freedom appearing in the general three-charge geometries. We will work in the D1-D5-P frame except for In the five-dimensional description, including the graviphoton, there are thus (n + 1) five-dimensional vector fields, A(I), encoded in the eleven-dimensional three-form potential C(3), and the scalars t(I), encoded in the Kahler form J for the compact six-dimensional C(3) = JI , J = JI . Here, JI are harmonic (1, 1)-forms on the compact six-dimensional space that are invariant under the projection performing the N = 2 truncation. In addition the t(I)s satisfy CIJK t(I) t(J) t(k) = 1 , where CIJK is given by the intersection product among the JI , so only n scalars are Here we will take the compact six-dimensional space to be T 6 . If we parametrize the T 6 by the holomorphic coordinates: w1 = x5 + ix6 , w2 = x7 + ix8 , w3 = x9 + ix10 , However, we will only need the subset of these: J1 2 J3 2 dw1 dw1 = dx5 dx6 , dw3 dw3 = dx9 dx10 , J4 (dw1 dw2 + dw1 dw2) = (dx5 dx7 + dx6 dx8) , J5 (dw1 dw2 dw1 dw2) = (dx5 dx8 dx6 dx7) . J2 2 dw2 dw2 = dx7 dx8 , In this basis, the only non-zero components of CIJK are C3JK CbJK = CbJK , J, K {1, 2, 4, 5} 0 0 0 1 our particular examples of a superstratum correspond to the presence of one extra vector t(5) = t(4) as in [48, 51]. In the STU model, the standard route (see, for example, the appendices in [52]) for getting from the IIB frame to the M-theory frame is to perform T-dualities on (x9, x5, x6) and then to uplift the resulting IIA description to eleven dimensions. In the IIB solution ratios correspond to the scalars in the vector multiplets. The fourth function represents a convenient way of writing the warp factor of the five-dimensional metric as a relaxation of the constraint (2.5): Z In our particular class of solutions with (2.8) and (2.9) we have Z3 = Z3 CbIJ ZI ZJ = Z3 Z1 Z2 Z42 = Z3P , as a warp factor in the six-dimensional formulation. The function, F , and the vector space components of the five-dimensional vector A(3), while the other two scalars Z1 and Z2 combine with a1 and a2 to give A(1) and A(2). As mentioned above, the degrees of freedom of the IIB solution (2.1) require an extra vector multiplet. In order to map the IIB configuration in the M-theory frame one needs a slightly more complicated combination of T-dualities and one S-duality [6]. However the final result is very similar to that of the STU model, with the scalar Z4 and vector a4 forming the new vector multiplet. in six dimensions. [53, 54]. This is more appropriate for our solution, since this formulation allows v-dependent solutions. Indeed, the six-dimensional formulation is the T 4 reduction of the IIB description in section 2.1. In the uplift, the five-dimensional graviton multiplet combines with one of the vector multiplets to yield the six-dimensional graviton multiplet, while all the remaining vector multiplets become anti-self-dual tensor multiplets. Thus the BPS equations for these systems were obtained in [21, 22] and were fully analyzed and greatly simplified in [23]. To build our solutions we need to add an extra anti-self-dual tensor multiplet and the corresponding analysis of the BPS equations is discussed in [27]. We now summarize this result and present the equations that we need to solve in order to construct a superstratum in the class of solutions presented in (2.1). The equations governing the supersymmetric solutions be a Killing vector. It is convenient to think of the fields in terms of the four-dimensional base geometry and so one defines a covariant exterior derivative F F 2 U , where a dot denotes differentiation with respect to v. It was shown in [23] that the supersymmetry constraints and the equations of motion have a linear structure and this will be crucial for the construction of solutions. The only intrinsically non-linear subset of constraints (the zeroth layer of the problem) is the restrict to a class of solutions where these constraints are trivially satisfied: we take the spatial base B to be R 4 and its metric ds24 to be the flat, v-independent metric. We will where 4 denotes the flat R 4 Hodge dual. but this means that some of the fields do not have a gauge invariant meaning. The field strengths can be written [27] in terms of the 2-forms Here and throughout the rest of the paper, d denotes the exterior differential on the spatial base B.7 The derivative, D, is covariant under diffeomorphisms mixing v and xi: v v V (xi) , Given that everything is u-independent, the class of diffeomorphisms of u that respect the form of the solution (2.1) may be recast in terms of a gauge invariance: 4 = Dx3 4 2 + a1 (D2 a4 d) + C . 7Note that this convention differs from that of much of the earlier literature in which the exterior differential on the spatial base B is denoted by d. The next layer (the first layer) of BPS equations determine the warp factors Z1, Z2, It is worth noting that the first equation in each set involves four component equations, while the second equation in each set is essentially an integrability condition for the first adding in the corresponding Zk yields four independent functional components upon which there are four constraints. and a second-order constraint that follows from the vv component of Einsteins equations9 = Z1Z2 +Z1Z2 +Z2Z1 (Z4)2 2Z4Z4 2 4 1 2 4 4 = v2(Z1Z2 Z42) (Z1Z2 (Z4)2) 2 4 1 2 4 4 . The important point is that these equations determine the complete solution and form a system that can be solved in a linear sequence, because the right-hand side of each equation is made of source terms that have been computed in the preceding layers of the BPS system. Outline of the construction of a superstratum We start in much the same way as in [1, 5, 12], with a round, D1-D5 supertube solution, in the decoupling limit. The geometry of this background is global AdS3 S3. The SU(2)L SU(2)R isometry of the S3 corresponds to the R-symmetry and the SL(2, R)L SL(2, R)R isometry of the AdS3 yield the finite left-moving and right-moving conformal groups. The mode analysis and holographic dictionary of this background is extremely well-understood [19, 20]. The background is dual to the Ramond ground state with max8Using the intersection numbers (2.9), the equations (2.19)(2.23) can be written more succinctly as 4DZI0 = CbIJ DJ , D 4 DZI0 = CbIJ I d, I = 4I , 4D 4 2 DF = 2 v(CbIJ ZI0 ZJ0 ) 2 b 1 CIJ ZI0 Z J0 4 CbIJ 4 (I J ), where Z10 Z1, Z20 Z2, Z40 = Z50 Z4, 4 = 5, and I 9This simplified form is completely equivalent to (2.9b) of [8]. quantities denote the right-moving sector counterparts). n1 and n5 are the number of D1 and D5-branes, respectively. The supertube shape modes associated with generic 14 -BPS spin of the underlying supergravity field. Thus, for a fixed spin field, these shape Fourier modes are determined by one quantum number and hence correspond to one-dimensional Adding momentum modes while maintaining 18 -supersymmetry means that we allow more general excitations in the left sector of the CFT in such a way that h > 0, while will generate intrinsically two-dimensional shape modes, for fixed spin. In this way, we can think of the superstratum as two-dimensional shape modes on the homology 3-cycle of the underlying microstate geometry. It is also useful to consider the NS sector states obtained by spectral flow from the and j = h. Acting on chiral primaries with SU(2)L SL(2, R)L generators generically the Ramond sector [55]. Since we know the action of SU(2)L SL(2, R)L on gravity fields, we can construct the modes corresponding to descendants10 of chiral primaries [8]. At the linearized level, we can take arbitrary linear combinations of these modes to make the superstratum. As we will see more explicitly in the next section, this will give us the solution of the first layer of the BPS equations. To construct the fully non-linear solution, we use the power of the observation [23] that the upper layers of the BPS equations are a linear system of equations. This means that the linear excitations can be used directly to obtain the complete solution in which the fluctuations are large. While simple, in principle, there are several essential technical obstacles to be overcome: (i) The construction of the generic linear modes explicitly in some manageable form. (ii) Solving the linear equations for the upper layers with sources constructed from combinations of the linearized modes. (iii) Removal of the singularities and building a smooth solution by fixing some of the Fourier modes but doing so in a manner that leaves two arbitrary quantum numbers, thus preserving the intrinsically two-dimensional form of the fluctuations. We now proceed to solve each of these problems one after another. This will mean that we have to dive into some very technical computations but we will regularly step back and orient the reader in terms of the goals stated here. 10The states obtained by acting R-symmetry generators on a chiral primary state must more precisely be called super-descendants, but for simplicity we refer to them as descendants. While supersymmetry does not allow the solutions to depend on u, states carrying momentum are generically going to be v-dependent. In the rest of this paper we will make the simplifying assumption that the four-dimensional metric ds24 is v-independent and simply along the D1-D5 common direction, is v-independent. We make this assumption simply for expediency; we do not know how to solve the system otherwise. These assumptions could, in principle, prevent us from finding a suitably generic superstratum because all the fluctuations that we will introduce in the other fields may ultimately require v-dependent superstrata will have v-dependence everywhere but our goal here is to demonstrate that there is at least one class of superstrata that is a suitably generic function of two variables. The fact that we will succeed despite this technical restriction is remarkable even though there are a posteriori explanations of this somewhat miraculous outcome. Two-charge solutions It is useful to think of the three-charge solutions as obtained by adding momentum-carrying perturbations to some two-charge seed. This will not only facilitate the CFT interpretation of the states but also give important clues for the construction of the geometries. All twocharge D1-D5 microstates have been constructed in [18, 20, 56, 57] and are associated with of the oscillating fundamental string dual to the D1-D5 system. The parameter along the curve is v0, which has a periodicity L = 2 QR5 where Q5 is the D5 charge and R is the In the duality frame of the fundamental string, the profile can be split into four R distinct role and, in fact, the D1-D5 geometries that have non-trivial values of gA(v0) for described by the class of solutions (2.1). We recall in appendix A how to generate the geometry from the profile gA(v0) for this restricted class of two-charge states. The simplest two-charge geometry is that of a round supertube, described by a circular profile in the (1, 2) plane: g1(v0) = a cos g2(v0) = a sin gA(v0) = 0 for A = 3, . . . , 8 . (3.1) The metric of the supertube is more easily expressed in the spheroidal, or two-centered, coordinates in which R4 is parameterized as these coordinates the flat R The metric coefficients specifying the supertube geometry are The parameter a is related to the D1 and D5 charges Q1, Q5 and the radius, R, of S1 by As one would expect, this geometry is asymptotic to R4,1 Q5 are related to the quantized D1, D5-brane numbers, n1 and n5, by the relation (A.3). The solution generating technique As usual, one can define a decoupling limit which corresponds to cutting off the asymptotic part of the geometry. This is achieved by taking (i = 1, 5) , and it implies that the 1 in the warp factors Z1 and Z2 can be neglected. In this limit, the supertube geometry reduces to AdS3 S3 T 4, as one can explicitly verify by performing the coordinate redefinition in the geometry (2.1a) with the data (3.4). Working in the decoupling region has the advantage that one can generate new solutions via the action of the symmetries of the CFT. These symmetries form a chiral algebra whose rigid limit is SU(2)L SU(2)R SL(2, R)L SL(2, R)R U(1)4. On the gravity side, each CFT transformation is realized by a diffeomorphism that is non-trivial at the AdS boundary. The SU(2) factors are R-symmetries of the CFT with generators {J0i, J0i}, erators {L0, L1, L0, L1}, are conformal transformations in AdS3. The U(1) factors are torus translations. The extension of these transformations to the full chiral algebra with extension of U(1) torus translations was considered in [10] and used to generate an exact family of three-charge solutions. Z1 = 1 + Z4 = F = 0 , Z2 = 1 + R = One can generate a three-charge solution by acting on a two-charge solution by a To preserve half (four supercharges) of the supersymmetry preserved by the two-charge state, one can only act with generators in the left-moving sector. For example, one can studied in [8], while the action on generic two-charge states at the linearized level was found a spectral flow (3.8) [55]. Explicitly, the relation is J+1 J1 = eS(J0+ J0)eS , where eS describes the coordinate transformation (3.8). The simplest two-charge geometry corresponding to the round profile (3.1) is mapped solution and we do not get a new three-charge solution. In order to generate a non-trivial three-charge solution, instead, one should start with a deformed two-charge seed. A rigidly-generated three-charge solution Perhaps the simplest two-charge seed solution12 that can be used to generate a new threegA. This produces a three-charge geometry that fits in the class (2.1), has undeformed g1(v0) = a cos g2(v0) = a sin g5(v0) = k where k is a positive integer and the remaining components of gA remain trivial. The corresponding two-charge geometry is described by Z1 = 2 Q5 F = 0 , Z2 = 11If we take the decoupling limit of the two-charge solution, the corresponding state in the boundary CFT is a ground state in the RR sector. By the level here, we mean the one in the RR sector. The momentum charge np is given by np = L0RR np = L0RR (modulo the zero-point energy shift by c/24). 12Another possibility is to turn on a density fluctuation on the profile (3.1) by changing the profile 1-forms and no Z4 would be generated, but Z1 and Z2 would be modified. L0RR. If one excites the left-moving sector only, this gives where we are restricting to the decoupling region and hence have dropped the 1 in Z1 and Z2. The relation between the parameters a, b, the asymptotic charges Q1, Q5, and the S1 radius R is now R = For fixed Q1, Q5, R and k, the solutions thus admit a freely varying parameter, that could be taken to be b/a. We will discuss in section 7 the CFT interpretation of this family of function Z2 remains unchanged. It is also very interesting to note that the combination in obtaining neutral black hole microstate geometries [59] and smooth coiffured black charge solutions (3.11) with nonzero b and obtaining a new three-charge solution13 [8]. The resulting solution represents a very particular three-charge state which, by construction, is a chiral algebra descendant of a two-charge state. We will refer to such a solution as a rigidlygenerated three-charge solution but we will use this solution as an inspiration to construct far more general classes of solution that are far from being rigid, and, in particular, are no rigidly-generated three-charge solution still has 13The explicit change of coordinates realizing e(J+1J1) on the gravity side can be found in [8]. As mentioned in section 2.3, we assume that the same happens in all three-charge geometries we consider, even if they are not descendant of two-charge microstates. So, hereafter, The Z4 in the rigidly-generated solution is a linear superposition of modes of the Z(k,m) = R r2 + a2 The solution also has a non-trivial spatial component of the NS-NS 2-form in (2.1d). One finds that this may be most simply written in terms of the gauge invariant quantities Note that these are not normalized but satisfy in the rigidly-generated solution. We will see in section 7 that this happens because the and thus should not appear in physically allowed solutions. Hence the modes, Z(k,m) and (k,m), are only allowed if m k. Note that, for these modes, the functions multiplying 4 rigidly-generated solution, the coefficients with which the terms Z(k,m) appear in the total Z4 are not all independent, but are fixed functions of a single parameter, the rotation A general class of solutions to the first layer The beauty of the solution generating technique is that it provides us with all the modes we need to solve the first layer of the BPS equations; indeed, one can explicitly check that each individual mode given by (3.15) and (3.17) solves the first layer of equations, (2.21). These modes depend upon two integers, (k, m), and provide an expansion basis for generic functions of two variables on the S3. So, as far as this layer of the problem is concerned, 14For an explicit example see appendix A of [8]. From that example one can also see that there exists one tions (2.22), (2.23) indeed imply that the change (6.3) necessarily generates a modification As usual one can choose a gauge in which F is unmodified. Finding the general solution to this equation is straightforward: if one takes the general r be done following a systematic procedure: one must first collect all the terms that give rise to them via (6.6) and (4.13). As before this will result in quadratics in the Fourier then solve for these Fourier coefficients and remove the singularities. We are not going to investigate asymptotically flat superstrata any further in this paper because it will take us into another rather technical discussion. We note that there are some simple examples of asymptotically-flat superstrata in [8] and we will leave the construction of families of asymptotically-flat superstrata to subsequent work [60]. Our primary focus for most of the rest of this paper will be the examination of CFT states that are holographically dual to the superstratum excitations and for this we only need the somewhat simpler classes of solutions in the decoupling limit, where we drop the 1s. For the purpose of computing the asymptotic charges of the solution, all the new terms arising from (6.3) and the concomitant cancellation of the singular modes are irrelevant because they are proportional to non-trivial Fourier modes in v and so vanish when integrated over the S1 compact direction. The angular momentum of the geometry can thus from the v-independent contributions generated by equal modes. The quantized angular momenta j and j are given by where V4 is the volume of T 4, gs is the string coupling and the dimension-full parameters J and J can be extracted from the large radius behavior of the geometry as: j = J , j = + O(r3) . For our solution we find J = k 1 J = Q1 = 1 X b2k,m k 1 Moreover the D1 supergravity charge can be extracted from the large distance behavior of the warp factor Z1 and is given by The D5 supergravity charge, Q5, is not affected by the superstratum fluctuations we consider. The integer numbers n1 and n5 of D1 and D5 branes are related to Q1 and Q5 by the relation (A.3). Altogether we find that the quantized angular momenta of our solution are j = N k 1 , j = N a2 . N k 1 A similar computation can be performed to derive the momentum charge of the solution. From the geometry we can extract the supergravity momentum charge Qp as Our geometry gives and hence its quantized momentum charge is 2 F = Qr2p + O(r3) . Qp = 1 X b2k,m k k 1 np = k 1 In particular, we note that from (6.11) and (6.15) we have np = j j . In the next section we will use the values of j, j and np to help determine the map between our geometries and the dual CFT states. The CFT description The geometries constructed in the previous sections are asymptotically AdS3 S3 According to the general AdS/CFT paradigm, we expect that they correspond to semiclassical states in the dual two-dimensional CFT, commonly called the D1-D5 CFT, with a large central charge c = 6N where N n1n5. contains 4 free bosons and 4 free fermions. Each circle in the figure corresponds to a single copy. (b) A twist field intertwines k copies into a single strand of length k. In this section we briefly recall the main features of the dual CFT that are relevant here17 and give a general description in the CFT language of the class of states dual to the superstratum geometries that we have constructed. Basic features of the dual CFT the non-compact R4 coordinates xi in the space transverse to all the branes. At a special point in its moduli space, this CFT can be described by a sigma model whose target space is the orbifold, (T 4)N /SN , where SN is the permutation group on N elements. Namely, we have N copies of 4 free compact bosons and 4 fermions, identified under permutations of Namely, under the R-symmetry, the bosons are singlets, while the left- and right-moving fermions transform as (2, 1) and (1, 2), respectively. It is useful to visualize the CFT states by representing the N copies, indexed by (r), as N strings (see figure 1(a)), on each of which live 4 bosons and 4 fermions: Besides the operators that can be built explicitly in terms of the free bosons and fermions, the CFT contains also twist fields that glue together k copies of the free fields into a single strand of length k. (See figure 1(b).) For this reason, the free point in this CFT moduli space is usually called the orbifold point. On a strand of length k, the k copies of the fields are cyclically glued together and of the fields on a strand of length k are 1/k times the mode numbers on a string of length 17For more details of the D1-D5 CFT, see e.g. [61] and references therein. one. General states have multiple strands of various lengths. For instance, if the `th strand nZ z k 2 . By construction, the excitations of the bosons, X(Ar A), only involve motions in the compactified (T 4) directions, whereas the fermionic excitations carry polarizations (Rcharge) that must be visible within the six-dimensional space-time. More concretely, the modes of the currents can be viewed as bosonizations of the fermions, and because these currents lie entirely in spatial directions of the six dimensional space-time it follows that suitably coherent excitations created by these currents will be visible within six-dimensional supergravity [5]. Note that one should not confuse the labels (r) and `: the former labels each set of bosons and fermions (7.2) before orbifolding whereas ` indexes the strands and so labels sets of bosons and fermions that have been orbifolded together to make a longer effective string. Thus the currents in (7.4) are defined for each individual strand labeled by ` and thus give a current algebra of level k, the length of the strand, rather than level 1, which would be the level of the current algebra for each individual set of fermions in (7.2). One can also write the current algebra of the R-symmetry by summing over all the fermions or over all the individual currents over all strands: 1 X 1 X X J`(z) , X J (z) . This current algebra has level N . The standard angular momentum operators, J i with i = 3, , are given in terms of the J by: J 3 = J 12 = J 21 , J + = J 11 , J = J 22 , and likewise for Ji. Also for the individual currents we similarly define J`i and J`i, from Even if the free description of the CFT lies outside the regime where supergravity is a reliable approximation, it is still a very valuable framework for describing the states dual to the superstratum. As usual, supersymmetry is responsible for this utility: the conformal dimensions of states preserving 1/8 of the total 32 supercharges and their 3point correlators [62] are protected and so, for these observables, it makes sense to match directly the CFT results obtained at the orbifold point to those derived in the supergravity A detailed comparison between these two pictures for the class of states described in this paper deserves a separate paper following the spirit of what was done 1. The circular profile given in (3.1) on the gravity side (shown on the left) corresponds to the CFT in [19, 20] for the 14 -BPS states. Here we will provide just the basic features of the duality between CFT states and bulk geometries. As one would expect, the first entry in the dictionary maps the global AdS3 S3 T 4 solution to the SL(2, C) invariant vacuum in the NS-NS sector. However, we are interested in states in the RR sector, which correspond to geometries that can be glued to an asymp T 4 region. The round supertube solution specified by the profile (3.1) is the simplest of such RR states. In order to find the CFT description for this state, it is sufficient to relate the change of variables (3.8) to the spectral flow on the CFT side. We first choose a U(1) U(1) subgroup of the R-symmetry group, and refer to the corresponding currents as J 3 and J3 (these currents correspond to the two U(1) rotation symmetries simply perform a spectral flow of the NS-NS vacuum state to the RR sector by using J 3 At the orbifold point, it is possible to write the J0i and J0i as the sum of generators acting Then, in the free CFT limit, the state is composed of N independent strands, each one with eigenvalues (j`, j`) = ( 12 , 12 ). This type of strands is annihilated by the modes (0+A )`. For a visual explanation of this correspondence, see figure 2. We can now build a dictionary between the supergravity solutions discussed earlier and a set of pure semi-classical states in the CFT. We parallel our approach to the supergravity solution by starting with a review of the 14 -BPS semi-classical states and their descendants. We then move to the CFT description of the fluctuating superstrata geometries by examining precisely how we added momentum modes to the 14 -BPS supertubes. 14 -BPS states and their descendants The semi-classical RR ground states that are dual to 14 -BPS geometries were discussed in detail in [19, 20] and here we will review the previous results in a language that is convenient for the generalization in the next section. All 14 -BPS geometries are determined by a closed profile gA(v0) in R8 but, as mentioned above, we focus only on a profile in an R 5 subspace in order to have states that are invariant under rotations of the T 4 coordinates. Thus, on the geometry side, we have to create superstrata. The profile given in (3.10) on the gravity side (shown on the left) corresponds to the CFT state with two types of strands (shown on the right). The first type of strand has length one and (j`, j`) = ( 1 , 1 ) while the second type of strand has length k and (j`, j`) = (0, 0). 2 2 using the language of the orbifold free field description, we can characterize the properties of the profile on the CFT side as follows: the mode numbers of the Fourier expansion profile determine the quantum numbers of each strand under the SU(2)L SU(2)R Rsymmetry generators, and finally the amplitude of each Fourier mode is related to the number of strands of a particular type present in the dual CFT state. Since we are focusing on 14 -BPS states, each strand has the lowest eigenvalue for both (L0)` and (L0)` and so the same is true for the full state. The profile in (3.10) represents a non-trivial deformation of the simple vacuum state represented by the profile (3.1) whose CFT interpretation, as discussed above, can be thought of as N strands of length 1. The profile (3.10) has an extra non-trivial component, g5, that has been added to the functions g1 and g2 that are already present in (3.1). It should therefore correspond to a state with two types of strands: the standard strands with and a second type of strand that is obtained from the first by acting with the operator ` AB . This operator is a scalar under rotations of the T 4 and carries are also invariant under R 4 rotations. Thus it is natural to associate these new strands to the component g5 of the profile. The coefficients a and b determine the number of the constituent strands of the first and the second type: a2 is proportional to the number of strands of the first type and b2/(2k) is proportional to the number of strands of the second type.18 Note that this is consistent with the relation (3.12), since the total length of the state is fixed in terms of N . Finally the Fourier mode numbers k of the various components of the profile determine the total length of the corresponding type of strands. We consider states for which the ( 12 , 12 ) strands have length 1, since this is the only Fourier mode present in g1 and g2 and the (0, 0) strands have arbitrary length, k.19 For a pictorial explanation of this correspondence, see figure 3. 18As discussed in [19, 20], this is not the exact characterization of the dual semi-classical state, even in the large n1n5 limit: in general the dual state is a linear combination of many terms, that is peaked around the configuration described in the text, with a spread determined by the coefficients a and b. 19Note that the geometries dual to these states do not have any conical defects even if the corresponding CFT state has strands of length k > 1. This is to be contrasted with the examples considered previously in the literature where all the components of the profile had the same Fourier mode k. nents of profiles gA with mode number k in gravity (shown on the left) correspond to the strands in the CFT states with length k with specific values of the R-charge (j`, j`) (shown on the right). In a similar way, it is possible to map different Fourier modes of each profile components to CFT strands with particular SU(2)L SU(2)R quantum numbers. The components g1ig2 of the profile correspond to ( 12 , 21 ) strands and the components g3ig4 correspond in the example above, this completes the dictionary between Fourier modes and strand types; see figure 4 for a visual explanation. Of course, supergravity solutions correspond to semi-classical states where each type of strand appears in many copies so as to be suitably coherent. The only relevant information for defining the dual state on the CFT side is the distribution of the numbers of each type of strand in the full state. Order one variations from the states discussed above are not visible within the supergravity limit. At this point it is straightforward to extend this correspondence to descendant states: both on the bulk and on the CFT side one just needs to act on the same 14 -BPS states with certain generators of the superconformal algebra. This programme was initiated in [55] and a general discussion at the linearized level can be can be found in [58]. In this paper have we focused on the R-symmetry generators. As summarized in section 3.3, a first example of a non-linear descendant geometry can be constructed simply by acting with the exponential by one unit and the average momentum of the descendant state is determined by the momentum expectation values between the bulk and the CFT descriptions). It is very important to understand the commonalities and differences between the construction of the rigidly-generated states obtained by the rotation above and our generic superstratum fluctuations. In the orbifold CFT language the rigidly-generated states contain not only the strands that were present in the original 14 -BPS states before the rotation but also have a new type of momentum-carrying strand that is obtained by acting with the superalgebra generators involved in the rotation. The relative number of the two types of strands (i.e. the RR ground states and the momentum-carrying ones) is determined by the rotation parameter. On the other hand, to make a fluctuating superstratum we rebuilt a complete supergravity solution starting from almost20 arbitrary superpositions of the linearized forms of all possible rigidly-generated states and thereby generated far richer families of CFT states. As we will see in the next subsection, the rigid rotation is crucial to developing the holographic dictionary for each individual mode and in this way we will obtain the CFT dual of the generic superstratum geometry. A class of superstrata: the CFT description We do not, yet, have an exhaustive description of the 18 -BPS geometries as we do for the 14 -BPS ones. So it is easier to construct the dictionary between supergravity solutions and semi-classical CFT states starting from the intuition built by studying the 14 -BPS solutions and working our way backwards. As described above, we use the orbifold point language and our proposal for the 18 -BPS dictionary is: A 18 -BPS solution in supergravity describing a finite fluctuation with modes vk,m given in (3.23) around AdS3 S3 corresponds in the CFT to a semi-classical state composed of strands of different types. The types of strands considered in this paper are characterized by the length, k, the (left-moving) momentum number, m, and the choice of fermion ground state ((0, 0) or ( 12 , 12 )). frequency with which each type of strand appears in the CFT state must be large, and corresponds in the bulk to how much the parameters of the supergravity Clearly the novelty as compared to the two-charge states is the appearance of a new quantum number m determining the momentum of each type of strand. In general, the momentum is carried by all possible types of excitation that are available in the CFT and, in particular, on a strand of length k, we can have modes of the free boson and fermion fields carrying a fractional quantum of momentum in units of the inverse of the radius R. The usual orbifold rules only constrain the total momentum on each strand to be integer-valued.21 For general momentum-carrying states on a strand, it seems quite non-trivial to determine the precise correspondence between the frequency with which the strand appears in the CFT state and the deformation parameter of the supergravity data. However, for the particular ansatz we consider here, the dictionary is simple enough and can be inferred from the data we have collected. As discussed in the gravity part of this paper, we have focussed on a class of states in 20Modulo the constraints imposed by regularity. 21Note however the important fact that this orbifold-CFT rule, that each strand carries integral units of momentum, must be refined in certain situations in the D1-D5 CFT; see [63, Section 6.3] for more detail. Here we ignore this point and only consider integral units of momentum on each strand. strands (recall that the subscript ` means that this operator belongs to the `th strand). This is the same type of strand that, in absence of momentum carrying excitations, is related to the g5 component of the profile (3.10). Thus it is natural to relate the presence of this type of strand to the presence of a term Z(k,m) (see (3.15)) in the supergravity solution. As a above for the 14 -BPS states: on the CFT side this kills the momentum-carrying excitations and on the gravity side we recover the solution (3.11) which can be obtained from the profile from (3.10) by using the general relations summarized in (A.1). It is now straightforward to characterize the states that are dual to the superstrata geometries we constructed: it is sufficient to look at the form of Z4 in (3.20) and interpret (j`, j`) = (0, 0) with m units of momentum carried by [(J+1)`]m/m!, with each term of the sum as indicating the presence of Nk,m strands (on average) of the type Nk,m = N k 1 (bk,m)2 N = We will also identify the number of strands of the type (j`, j`) = ( 1 , 1 ) with N a2. The 2 2 numerical factors are suggested by the supergravity expressions for the charges derived in section 6. First, in our superstratum state the average numbers of strands of winding k multiplied by k should sum up to the total number of CFT copies: N = N a2 + k 1 (bk,m)2 This relation matches (6.10). Also the angular momentum charges (6.11) can be easily checked from the microscopic picture: only the first type of strands in figure 5 carries rightmoving angular momentum. Since the number of such strands is proportional to a2, this matches the second relation in (6.11). The first relation in this equation follows from the where it acts. Indeed, (6.16) shows that one quantum of momentum is associated with one The identification between each single term Z(k,m) and the presence of many copies of an excited type of strand is also supported by some general properties of the momentumcarrying operator we used. For instance, by using the free orbifold description of the CFT it is possible to see that [(J+1)`] m vanishes when it acts on a strand of length k if m > k. This can be easily verified in the orbifold CFT. One can simply note that this is a standard null vector identity in a current algebra of level k, or one can use (7.3) and (7.4) ( +n1 )`(1+2 nk )` with 0 n k; so in [(J+1)`]m with m > k at least one fermionic creation can see from equation (3.16) exactly the same constraint arises on the supergravity side as CFT on the strand. The wavy arrows represent the non-vanishing momentum modes excited on the strand. The number of strands of the same type is O(N ), meaning that our solution represents a finite non-linear deformation around the AdS3 S3 background. For the precise numbers of each type of strand, see (7.8). Another check that one can perform is to choose very particular values for the parameters defining the superstratum states so as to reconstruct a descendant state. For instance, in figure 5 we can consider just a single type of momentum-carrying strand with of momentum-carrying strands has to be equal to the (average) number of (0, 0) strands in The main message of this construction is that the linearized (in the parameters bk,m) expression for the scalar fields Zi is sufficient to identify the dual state on the CFT side. On the bulk side the supergravity equations allow one (at least for this class of states) to find the explicit non-linear solutions. At this point it is possible to further check the dictionary between CFT states and geometries by comparing the expectation values for the protected operators as it was done for the 14 -BPS solutions in [19, 20] and for their 18 -BPS descendants in [27]. To conclude this section we want to underline the significance of the three-charge supergravity solutions that we have built. The three-charge geometries with a precise CFT dual that have been known prior to this paper [8, 10, 14, 15] have been obtained by a solution-generating technique [55] that amounts to applying R-symmetry generators class of momentum-carrying states. In technical terms, one can only obtain the R-current descendants of chiral primaries.22 In contrast, our geometries correspond to descendants of non-chiral primaries and specifically, states generated by the small current algebras, (7.4), on different types of strand. Our approach thus yields completely new, broad classes From the CFT perspective, the way we have achieved this can be described more precisely as follows: Our three-charge states, such as the one described in figure 5, are 22There is minor abuse of terminology here: since the D1-D5 CFT is in the RR sector, what we really mean by a chiral primary is the spectral flow of a chiral primary in the NS-NS sector. The same caveat applies to the subsequent discussions. as representing a descendant of a chiral primary. However, when we have two or more strands, the full state is a tensor product of descendants of chiral primaries. Now recall that, although the tensor product of chiral primaries is again a chiral primary, the tensor product of descendants of chiral primaries is in general a descendant of a non-chiral primary [64]. Therefore, a multi-strand state in general represents a descendant of a non-chiral primary. for special states in which numbers of different types of strand are tuned in some precise way. We can see the same physics from the supergravity perspective: in our solution we allowed for arbitrary coefficients bk,m in the linear combination in (3.20). This means that, in general, it will not be possible to rewrite the solution as an element of the R-symmetry group, which is an exponential of the operator J i = P`(J i)`, acting on a two-charge discussed in subsection 3.3 or those considered in [8, 27] appear as special cases where the coefficients of the various terms in the expression for the ZI are chosen in a precise way that allows one to reconstruct the currents J i. Discussion, conclusions and outlook First and foremost we have constructed an example of a superstratum with sufficient genericity to substantiate the claim that the superstratum exists within supergravity as a smooth solution parameterized by at least one function of two variables. This, in itself, represents huge progress within the programme of reproducing the black-hole entropy by counting microstate geometries that are valid in the same regime of parameters where the classical black hole solution exists, and is cause enough for the white smoke and celebration suggested by this papers title. At a more technical level we have given an algorithm that can be effectively implemented to generate shape modes of the superstratum. We have also begun to develop a systematic picture of the holographic duals of our superstrata and the results presented here contain several important new results: up until now, all the three-charge geometries constructed by solution generating methods starting from two-centered geometries [8, 10, 14, 15] were dual to descendants of chiral primary states23 in the left-moving sector of the D1-D5 CFT. To obtain the most general 18 -BPS state one must be able to find the gravity duals of arbitrary left-moving states: descendants of non-chiral primaries. We have shown how such states are indeed being captured by the Our focus in this paper has been to exhibit one example of a superstratum rather than attempt an analysis of the possible families of superstrata. As a result we have passed over many interesting and important physical and mathematical issues that arise from our construction and we would like to catalog some of them. We begin with the interpretation of our work in terms of the CFT. In [5], three of the present authors conjectured that the fluctuations of the superstratum that are visible in six-dimensional supergravity correspond to current algebra excitations of the CFT. The 23See footnote 22. current algebra in question is generated by the modes (J ni)` of (7.4) acting on individual which is large enough to reproduce the asymptotic growth of the entropy of the threecharge black hole. This is in stark contrast to the R-symmetry algebra generated by the understand the fluctuation modes of the superstratum and reproduce the black-hole entropy growth, it is crucial to study how individual generators (J ni)` are realized in supergravity and whether they generate smooth geometries. The solution generating technique that was used in the literature [8, 10, 14, 15] to obtain smooth three-charge solutions starting from two-center geometries constructs solutions that are descendants of a chiral primary by the action of the total generator J ni, and thus does not allow one to change independently each individual (J ni)`. However, by taking a tensor product of such descendant states, which corresponds in supergravity to taking a linear superposition24 and non-linearly completing it, we successfully constructed smooth momentum-carrying geometries dual to states that are not the result of acting on chiral primaries with the total J ni but intrinsically involve individual generators (J+1)`. These explicit solutions demonstrate that the action of some individual generators are indeed realized as smooth geometries. We regard this as evidence in support of the conjecture generated by the individual generators (J ni)`. Furthermore, the fact that our solutions involve two parameters k, m suggests that the general fluctuation of the superstratum is described by functions of at least two variables, as claimed in [5]. Although this represents major progress toward showing that the action of the full algebra of individual generators (J ni)` gives smooth superstrata in gravity, there are still black hole entropy growth S n1n5np if np n1n5, while on a strand of length k = N , fractional modes can account for the entropy growth if n1n5np 1. Therefore, either higher modes or fractional modes are separately sufficient to reproduce the three-charge black hole entropy for large enough np. We can look at these issues with the modes (J ni)` from a different angle. As we have argued, the smooth geometries we have constructed represent the tensor product of the interpretation as states of a supergraviton [65]. When there are multiple supergravitons, the total state is the tensor product of such supergraviton states. Therefore, the smooth 24This superposition can be understood as follows. If one has a free harmonic oscillator with the annihilaIf one has two oscillators with a and b, the classical configuration in which the a oscillator with amplitude geometries constructed in this paper must correspond to the states of a supergraviton gas in the bulk. More precisely, our geometries can be regarded as coherent states in the multiparticle Hilbert space of supergravitons. Conversely, we expect that the quantization of our solutions reproduce the multi-particle supergraviton Hilbert space around AdS3 S3.25 It was shown in [66, 67] that the supergravity elliptic genus computed by counting these supergravitons, with a stringy exclusion principle imposed by hand, agrees with the N4+1 , or in the R sector, L0R J 3 + 14 . (See figure 6.) In other words, in this parameter region, it has been shown that every CFT state has a bulk realization as a multi-supergraviton state, modulo the fact that some states are missed because elliptic genus counts states with signs (it is only an index). Therefore, our geometries must be giving the bulk semi-classical description of all CFT states in this parameter region (again, modulo possibly missed states). However, this observation also illuminates what states our solutions fail to capture. The results in [66, 67] imply that, CFT states. By construction, the supergraviton gas includes neither higher nor fractional modes and so we need these modes to reproduce the entropy above the bound. In particular, because single-center supersymmetric black holes (the BMPV black hole [68]) exist above the bound, we certainly need to understand superstratum realizations of higher and/or fractional modes to reproduce entropy of this black hole. Actually, the story is even more interesting, since in [69] it was shown that there are multi-center black holes (moulting black holes) even below the bound. These black holes must correspond to higher and/or fractional modes that are not visible in the elliptic genus because of cancellations between bosonic and fermionic states. Therefore, understanding superstratum realizations of higher and/or fractional modes are important also for understanding the microstates of moulting black hole configurations. The microstates of the moulting black holes may be promising for studying higher and fractional modes, because we can study those modes by looking at small deformations around AdS3 S3. Returning to the supergravity perspective, some important new ingredients are needed to make further progress in the construction of the most general supergravity superstrata. As we noted above, the solutions constructed in this paper are based on the action of factor ei 2 nv/R [58]. In general, the action of the total Jn makes all kinds of quantities i particular, the new parameter, n, introduced by this procedure is expected to generalize 0i with a v-dependent exponential the phase vk,m in (3.23) to vk,m,n depending on three parameters and lead to a much broader class of three-charge solutions. which we did not consider in this paper. To reproduce the full Hilbert space, one needs to include the bulk geometries generated by these generators too. CFT elliptic genus can be reproduced by the bulk graviton gas for LR Figure 6. The J 3-L0R phase diagram of the D1-D5 system. Pure AdS3 S3 corresponds to the hatched region). Single-center BMPV black holes exist for L0R region). Even in the region L0R < (J3)2 + N4 , there exist multi-center configurations of black holes N 0 J 3 + 14 (blue horizontallyand rings with a finite horizon area [69]. Furthermore, much of the three-charge entropy comes from excitations on strands with the gravity dual, excitations with this energy gap are known to come from fluctuations of deep, scaling geometries in which the wavelength of the fluctuation is approximately the scale of the horizon [26, 7072]. There are thus two ways we might find such superstrata: one could consider a single, large superstratum with a large dipole moment, k, and hence a very large order, Zk, orbifold singularity and then allow multi-valued functions with the fluctuation spectrum.27 (Of course, multi-valued excitations are not allowed in supergravity and therefore they must be excited multiple times so that their wavefunction is single While such a configuration is technically singular, its physical meaning is still understandable. Alternatively, one could completely bubble such a configuration to a kcentered configuration and then the lowest energy fluctuation will be some collective mode of all the bubbles in this configuration in some deep scaling limit. While the latter would 26BPS excitations can only have a gap of order O(1), because supersymmetry means that the excitation energy is equal to the momentum number which is quantized to integers. Non-BPS excitations, on the other hand, are not subject to such constraints and their energy gap can be as small as O(1/N ). 27A similar approach was used in [73] to construct a restricted class of microstates containing frachave no orbifold singularities, its lack of symmetry might make analytical computations More generally, there remains an important conceptual issue in microstate geometries: in the deep, scaling holographic dual geometries, but a detailed understanding of precisely how these dual states are related remains unknown. As indicated above, part of the story must involve resolving orbifold singularities and multi-valued functions but, on the gravity side, it must also involve deep scaling geometries. We would very much like to understand the emergence of such scaling geometries from the detailed matching in the holographic dictionary. Understanding this issue is going to be an essential part of seeing how the CFT entropy is encoded in the bulk geometry. In the construction presented here we also encountered new types of singularities that are more difficult to remove than the singularities that appear in the standard construction of five-dimensional microstate geometries. In the latter, the removal of singularities was related to the removal of closed time-like curves and this could be achieved by adjusting the choices of homogeneous solutions to the linear system of equations underlying the BPS solutions. Here we have found that the choice of homogeneous solutions is insufficient for the task of singularity removal: one also has to interrelate the otherwise independent sets of fluctuations in order to obtain non-singular solutions. We also noted that these interrelationships are very similar to those required for smooth horizons in black rings with fluctuating charge densities [29]. The physical origins and the resolutions of these potential singularities remains unclear and in this paper we simply exploited a mathematical algorithm to remove such singularities. We would like to understand the origins of such singularities, classify the ways in which one can cancel them and see if there is, indeed, some physical link of this to black-ring horizon smoothness. In this paper we have also focused on superstrata that are asymptotic to AdS3 S3. This choice was made for two reasons: simplicity and holography. The removal of singularities is simpler if the space is asymptotic to AdS3 S3 and such asymptotics is all one needs for the study of the states that are holographically dual to our superstrata. More generally, we would like to construct and classify superstrata that are asymptotic to T 2. This means that constant terms need to be introduced into some of the metric functions. As noted in section 6, the constructions of such solutions should involve only straightforward technical issues rather than serious conceptual or physical issues. Indeed, such solutions will be investigated in [60]. There are also some other interesting technical issues in the mathematics of superstrata. First, we found in sections 4 and 5 that regularity required our Fourier coefficients to satisfy a quadratic constraint and that constraint came from canceling a class of terms appearing in the quadratic Z1Z2 Z2. Motivated by solution generating methods [8, 10, 14, 15] we 4 constraints and hence more allowed excitations of the superstratum. As we also noted, one must furthermore revisit the quadratic constraint on Fourier coefficients if one is to construct superstrata in asymptotically flat geometries and so we intend to analyze this constraint more completely in [60]. The second technical issue has to do with the existence of a systematic approach to solving the system of differential equations underlying our solutions. We have been in this paper able to completely solve for all the fields of the solution in closed form except for one function appearing in some of the components of the angular momentum vector.28 The two-centered system leads to some relatively simple differential operators and, in particular, the Laplacian (4.9) is separable. The sources are also relatively simple functions and we have managed to find complete analytic solutions for some infinite families of sources. Our that there must be a far more systematic approach to solving this system of differential equations. Indeed, we strongly suspect that the whole mathematical problem we have been solving in section 4 should have a much simpler formulation and solution in terms of some cleverly chosen orthogonal polynomials. Understanding this may, in turn, lead to a clearer understanding of the whole system of differential equations and maybe even a reformulation of the general solution, perhaps even for multi-centered solutions, in terms of Green functions. Furthermore, solving this problem should enable the complete analytic construction of the most general superstratum based on two centers. Work on this is We would like to thank Jan de Boer and Samir Mathur for discussions. The work of IB was supported in part by the ERC Starting Grant 240210 String-QCD-BH, by the National Science Foundation Grant No. PHYS-1066293 (via the hospitality of the Aspen Center for Physics) by the John Templeton Foundation Grant 48222 and by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-RFP3-1321 (this grant was administered by Theiss Research). The work of SG was supported in part by the Padua University Project CPDA144437. The work of RR was partially supported by the Science and Technology Facilities Council Consolidated Grant ST/L000415/1 String theory, gauge theory & duality. The work of MS was supported in part by Grant-in-Aid for Young Scientists (B) 24740159 from the Japan Society for the Promotion of Science (JSPS). The work of NPW was supported in part by the DOE grant DE-SC0011687. SG, RR and NPW would like to thank Yukawa Institute for Theoretical Physics for hospitality at the Exotic Structures of Spacetime workshop (YITP-T-13-07) during the early stages of this project. SG, RR, MS and NPW are very grateful to the IPhT, CEA-Saclay for hospitality while a substantial part of this work was done. MS would like to thank the high energy theory group of the University of Padua where this work was completed for hospitality. Z2 = 1 + Z1 = 1 + A = L |xi gi(v0)|2 Z4 = L dB = 4 dA , F = 0 , ds42 = dxidxi , a1 = a4 = x3 = 0 , the dual with respect to the flat R 4 metric ds24 = dxidxi. The D1 charge is given by The quantities Q1, Q5 are related to quantized D1 and D5 numbers n1, n5 by Q1 = Q1 = The 14 -BPS D1-D5 geometries invariant under T 4 rotations are associated with a profile the functions and fields describing the geometry in the language of the IIB solution (2.1) are (p,q)Gkm = [p2 (k + 2)2]Sk+2,m+2 + [(k m)2 (p q)2]Sk,m+2 + (m2 q2)Sk,m. where V4 is the coordinate volume of T 4. Solution of the (generalized) Poisson equation The function Fk(,pm,q) was defined in the main text to be the regular solution to equation (4.18), which we repeat here for convenience: (p,q)Fk(,pm,q) = where the generalized Laplacian Lb the explicit solution to this equation, given in the main text in equation (4.20). (p,q) was defined in (4.8). In this appendix, we derive First, let us define the functions It is easy to show that these satisfy the following recursion relation: Now, let us introduce the following generating functions: In terms of these, the equation we want to solve, (B.1), can be collectively written as = 1 e2 ( + 2)2 (p q)2 = and the recursion relation (B.3) as Lb(p,q)G(, ) = he22(p2 2)+e2(( + 2)2 (p q)2)+(2 q2)iS(, ). F (, ) = he22(p2 2) + e2(( + 2)2 (p q)2) + (2 q ) 2 i1 k 2p s1 k 2p 1 k1s1 m1t1 mk1111 k2s1 mk2211 By expanding the nth power using binomial coefficients and explicitly writing down the first few terms in terms of Fk(,pm,q) and Gk,m, one finds Fk(,pm,q) = X X s=0 t=0 s (k m + p q)(k }| m + p q 2) { (zm + q 2)(}m| + q 4) { z st (k + p)(k + p 2) st k+2p s1 t1 k+2p 1 m2+q 1 (k p)(k p 2) where the relations between (k, m, p, q) and (k1, m1, k2, m2) are given in (4.11), (4.15). 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Iosif Bena, Stefano Giusto, Rodolfo Russo, Masaki Shigemori, Nicholas P. Warner. Habemus superstratum! A constructive proof of the existence of superstrata, Journal of High Energy Physics, 2015, 110, DOI: 10.1007/JHEP05(2015)110