Integrability and black-hole microstate geometries

Journal of High Energy Physics, Nov 2017

Iosif Bena, David Turton, Robert Walker, Nicholas P. Warner

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Integrability and black-hole microstate geometries

HJE Integrability and black-hole microstate geometries Iosif Bena 0 1 2 5 David Turton 0 1 2 5 Robert Walker 0 1 2 3 Nicholas P. Warner 0 1 2 3 4 0 Los Angeles , CA 90089 , U.S.A 1 Los Angeles , CA 90089-0484 , U.S.A 2 Orme des Merisiers , F-91191 Gif sur Yvette , France 3 Department of Physics and Astronomy, University of Southern California , USA 4 Department of Mathematics, University of Southern California , USA 5 Institut de Physique Theorique, Universite Paris Saclay, CEA , CNRS We examine some recently-constructed families of asymptotically-AdS3 supergravity solutions that have the same charges and mass as supersymmetric D1-D5P black holes, but that cap o smoothly with no horizon. These solutions, known as Black Holes in String Theory; AdS-CFT Correspondence - 3 superstrata, are quite complicated, however we show that, for an in nite family of solutions, the null geodesic problem is completely integrable, due to the existence of a non-trivial conformal Killing tensor that provides a quadratic conservation law for null geodesics. This implies that the massless scalar wave equation is separable. For another in nite family of solutions, we nd that there is a non-trivial conformal Killing tensor only when the left-moving angular momentum of the massless scalar is zero. We also show that, for both these families, the metric degrees of freedom have the form they would take if they arose from a consistent truncation on S3 down to a (2 + 1)-dimensional space-time. We discuss some of the broader consequences of these special properties for the physics of these black-hole microstate geometries. 1 Introduction 2 3 4 5 1 3.1 3.2 4.1 The (1; 0; n) family of solutions 4.1.1 4.1.2 4.1.3 The special families of superstrata metrics The last year has seen a signi cant breakthrough in the construction of microstate geometries [1, 2]. In particular, microstate geometries corresponding to ve-dimensional, three-charge, supersymmetric black holes with arbitrarily small angular momenta have been constructed. These solutions are horizonless and smooth and have an arbitrarily-long BTZ-like throat that interpolates between an AdS3 S3 asymptotic region and a long very-near-horizon AdS2 region. Deep inside the AdS2 region, the throat caps o smoothly just above where the black-hole horizon would be [1]. In the D1-D5-P frame these microstate geometries, known as \superstrata" [ 3, 4 ], have been proposed as holographic duals of speci c families of pure states of the D1-D5 CFT, involving particular left-moving momentum-carrying excitations [1],1 with charges in the regime of parameters in which a large BPS black hole exists. Hence these solutions correspond to microstates of a black hole with a macroscopic horizon. The momentum excitations may be thought of as creating the long AdS2 black-hole-like throat; in the full solution these momentum excitations are located deep inside that throat, and support its macroscopic size. As one descends the AdS2 throat, to an excellent approximation it is almost identical to a black hole throat until near the bottom, where one encounters the momentum excitations, before the geometry caps o smoothly. Given that we have a holographic understanding of these solutions, and that the proposed dual CFT states live in the same ensemble as the states that give rise to the black 1A subset of these microstate geometries can be mapped to excitations of the MSW string that carry momentum and angular momentum, via a sequence of solution-generating transformations and string dualities [2]. { 1 { e.g. [5, 6]), and should also give rise to novel physics where the thermodynamic description breaks down. Hence, we would like to understand how, for example, horizonless geometries scatter and absorb incoming particles, and how this di ers from the classical black hole result. The states we will study are still somewhat atypical, and will have interesting differences from the corresponding classical black hole; we hope the present study will inform future studies of progressively more typical microstates. Superstratum solutions are parameterized by arbitrary functions of (at least) two variables [4]. Generic superstratum solutions depend on all but one of the coordinates in six dimensions, and hence may appear complicated and somewhat intimidating to the uninitiated. However, in this paper we will show that two particular asymptotically-AdS3 families, each parameterized by one positive integer, have much simpler physics than may S 3 have been expected. One of these families, which we will call the (1; 0; n) family, has a separable wave equation and a conformal Killing tensor.2 This implies that the equations for null geodesics in these geometries are completely integrable: there is a complete set of conserved quantities, that are linear or quadratic in velocities. Related work on geodesic integrability in (two-charge) black hole microstate geometries and D-brane metrics can be found in [7, 8]. We will also show that the metrics of the (1; 0; n) family, and of another family that we will call the (2; 1; n) family, can be re-written in a form that they would take if they arose from consistent truncations on S3 to (2 + 1)-dimensions. By this, we mean the following: we can write each metric in terms of a (deformed) S3 bration over a three-dimensional base, K, where the metric on K depends only on the coordinates on K. Moreover, the bration and warp factors conform to the standard KK Ansatz for vector elds and Einstein gravity on K. In this paper we will restrict our attention to the metric degrees of freedom, and we postpone a full analysis of the existence, or otherwise, of a complete consistent truncation to future work. Such an investigation would require the inclusion of the six-dimensional tensor gauge elds in the consistent truncation Ansatz, building upon the results of [9, 10] to include more six-dimensional tensor multiplets. The purpose of the present work is to elucidate the metric structure and how it takes the form it would take if it came from a consistent truncation, since this is a remarkably strong constraint upon its structure. The metrics of the (1; 0; n) and (2; 1; n) families also have several isometries, such that the reduced metric on K, and the KK elds of the bration, depend only on one coordinate (which asymptotically becomes the radial coordinate of the AdS3). Thus, the analysis of these metrics can, in principle, be carried out entirely using three-dimensional gravity coupled to vector elds. 2This provides a quadratic conserved quantity for null geodesics. { 2 { Unlike in the (1; 0; n) family, the wave equation is not generically separable in the (2; 1; n) family, and the general geodesic problem does not appear to be completely integrable. However, these microstate geometries come very close to having these properties: for waves or geodesics that have zero SU(2)L angular momentum on the S3, one does have separability of the wave equation and complete integrability of the null geodesic equations. As we will discuss, complete integrability of geodesics may be both a blessing and a curse. In particular, completely integrable systems have highly restricted spectra and limited scattering behavior, and thus may not reveal some of the interesting physics of generic microstate geometries. The complete set of conservation laws lead us to suspect that the (1; 0; n) geometries will quickly eject infalling particles, and hence will not reproduce the expected black-hole thermodynamic behavior. On the other hand, the absence of integrability in the (2; 1; n) family will allow for more complicated dynamics, and possibly the trapping of incoming particles, which is a step closer to the behavior one expects from typical microstates. In a sense, the (2; 1; n) family provides an ideal setting for investigating more complicated dynamics: one can probe the behavior of the non-integrable geodesics by doing perturbation theory for waves and geodesics about the integrable (jL = 0) ones. Thus one can probe quite non-trivial scattering properties in a controlled expansion. There are three recent, seemingly unrelated, lines of investigation to which our results should be relevant. The rst is an argument that supersymmetric microstate geometries should exhibit a non-linear instability [ 11 ] (see also [12{14]). This proposed non-linear instability is related to the existence of stably trapped null geodesics deep inside the core of microstate geometries, however so far this has only been explicitly analyzed for very symmetric geometries, none of which have charges and angular momenta corresponding to a black hole with a large horizon area. Our results should help elucidate whether or not this proposed non-linear instability is an artifact of very symmetric microstate geometries. The second line of investigation is the late-time behavior of correlation functions in the D1-D5 CFT, and its connection to quantum chaos [15]. One expects chaotic systems to exhibit late-time uctuations that come from their underlying microscopic description, and that are not visible in the thermodynamic approximation. Hence, one expects typical bulk microstates to give rise to late-time uctuations that are not visible in the classical black-hole solution. For a set of two-charge black hole microstates these uctuations have been computed in the dual CFT [16, 17], but the corresponding computations in the bulk are beyond the capability of present technology. We hope that our results will open the way for a new testing ground for these questions. The third line of investigation is the computation of four-point functions of two heavy and two light operators; for a few examples, see [18{24]. Such calculations can be done in the CFT and can be matched to the light-light two-point function computed holographically in the microstate geometry dual to the heavy state [22, 23]. So far, the bulk calculation has only been done in microstate geometries that are dual to very special heavy states (a particular set of R-R ground states and spectral ows thereof [ 25 ]), essentially because of the technical di culty of solving the wave equation. The integrability of geodesics and the separability of wave operators in the in nite families of microstate geometries we consider should simplify the calculation of the two-point functions and vastly enhance the number { 3 { of two-heavy-two-light four-point functions that can be computed in the bulk. Moreover, since there is an explicit proposal for the CFT states dual to the microstate geometries we study [1], these four-point functions could be compared to those computed in the CFT. Furthermore, since our geometries can have long black-hole-like AdS2 throats and closely resemble black holes far from the cap region, these four-point functions should shed light on how unitarity is restored when replacing the black hole horizon with a fuzzball. This paper is organized as follows. In section 2 we describe in more detail the special properties of the metrics in the (1; 0; n) and (2; 1; n) families of microstate geometries. In section 3 we give a brief description of these families of BPS solutions. In section 4 we give the details of the metrics of the (1; 0; n) and (2; 1; n) families, separate variables in the wave equation, and describe the conformal Killing tensor. Finally, in section 5 we discuss the important features of the geometries described in this paper, and further discuss the implications of our results. 2 Dimensional reduction, separability and Killing tensors We work in Type IIB string theory on M4;1 S 1 M, where M is either T4 or K3. The circle, S1, is taken to be macroscopic and is parameterized by the coordinate y, with radius Ry: y y + 2 Ry : We consider a bound state of D1-branes wrapped on S1, D5-branes wrapped on S1 and momentum P along S1. The internal manifold, M, is taken to be microscopic, and we assume that all elds are independent of M. Upon dimensional reduction on M, one obtains a theory whose low-energy limit is six-dimensional N = 1 supergravity coupled to two (anti-self-dual) tensor multiplets. This theory contains all elds expected from the study of D1-D5-P string world-sheet amplitudes [26]. The system of equations describing all 18 -BPS solutions of this theory was found in [27]; it is a generalization of the system discussed in [28, 29]. Most importantly, this BPS system can be greatly simpli ed, and largely linearized [30]. For supersymmetric solutions the six-dimensional metric is well known to take the form [28]: (2.1) M, (2.2) (2.3) 3 ds62 = 2 p P where we take 1 2 u = p (t y) ; v = p (t + y) : 1 2 Supersymmetry requires that all elds be independent of u, but generic supersymmetric solutions can depend upon all the other coordinates. Upon taking the AdS/CFT decoupling limit [31], one obtains asymptotically AdS3 S solutions. We will work exclusively in the decoupling limit throughout this paper. We shall study solutions whose tensor elds have explicit dependence on v, as well as on the S3. These solutions are known as \superstrata" [1{4, 32]. In the solutions we study, the metric, (dv + ) du + ! + 1 2 F (dv + ) + p P ds42 (B) gMN dzM dzN ; { 4 { by (x1; x2; x3) ds62 At in nity, the sets of coordinates (u; v; r) and ( ; '1; '2) parametrize AdS3 and S3 respectively. The superstratum solutions that we consider were constructed in [1], and they have the property that the tensor elds depend explicitly on a single linear combination of v, '1 and '2. We thus refer to them as single-mode superstrata. In these asymptotically AdS3 S3 solutions, this phase-dependence cancels in the energy-momentum tensor, and hence in the metric.3 Thus the metric has isometries not just along u (as required by supersymmetry) but also along v, '1 and '2. In this paper we shall exploit these enhanced symmetries and examine the remaining, highly non-trivial dependence on (r; ). Our rst goal is to re-write the general six-dimensional metric as a bration of the compact three-manifold, S, described by (y1; y2; y3) ( ; '1; '2), over a base, K, parametrized (u; v; r). Speci cally, we re-cast (2.2) in the following form: gMN dzM dzN = 2 g^ dx dx + hij (dyi + Bi dx )(dyj + B j dx ) ; (2.6) where g^ and hij are viewed as metrics on K and S respectively, and where to be the volume of S divided by the volume of the S3 to which S limits at in nity: is de ned pdet(hij ) pdet(hij )jr!1 : In a general BPS solution, g^ , hij , Bi and It is convenient to de ne the metrics: ds12;2 g^ dx dx ; ds32 hij dyidyj : At in nity, ds21;2 is asymptotic to the metric on AdS3, and ds23 is asymptotic to the metric on S3. It is also useful to observe that one can invert the form of gMN in (2.6) explicitly: can depend on all the coordinates, except u. gMN = gMN = 2 2 g^ + hkm Bk Bm hik Bk Bk hkj hij g^ Bi g^ g^ Bj 2hij + g^ Bi Bj ! ; ! : ds24, on the four-dimensional base, B, is at and we write it in the standard bipolar form: dr2 combination of the vector elds, Bi , and the metrics g^ and hij . In particular, we note that the inverse metric on the internal space, S, is a non-trivial The warp factor, 2, in front of ds21;2 in (2.6) is precisely the factor needed for the dimensional reduction from six dimensions down to the three-dimensional space time, K. To be more speci c, this is the warp factor needed to reduce the six-dimensional Einstein action down to the three-dimensional Einstein action for g^ on K. In general, attempting to perform such a dimensional reduction is of course not very useful, because g^ in (2.6) will typically depend upon the yi. 3Upon completing these solutions to asymptotically R1;4 S1 solutions, the metric depends explicitly on the linear combination of v, '1 and '2 [33]. { 5 { (2.4) (2.5) (2.7) (2.8) (2.9) However, the (1; 0; n) and (2; 1; n) single-mode families of BPS geometries both have the property that: (i) The metric g^ is only a function of r. Moreover, the (1; 0; n) family also has the following remarkable feature: (ii) On ds26, the massless wave equation and the Hamilton-Jacobi equation for null geodesics are separable. The \massive" wave equation on ds26 is also separable if the mass term is induced from a mass as seen by the (2+1)-dimensional metric ds21;2. Finally, it is elementary to verify that the following is true for all metrics of the form (2.2): (iii) If one computes p along the base de ned by (r; ; '1; '2) are identical to the components of p g~ g~ab, g gMN for the six-dimensional metric, then the components of this where g~ab is the metric de ned in (2.4). Property (i), combined with equations (2.6) and (2.7), means that the complete sixdimensional metric has the form it would take if it arose from a consistent truncation to three-dimensional physics on K. This three-dimensional geometry is furthermore determined entirely by functions of r alone. Property (ii), the separability of the massless wave equation and of the massless Hamilton-Jacobi equation for null geodesics, implies the existence of a \hidden symmetry": there is another quadratic conserved quantity for the null geodesic equation [34, 35].4 That is, in addition to the usual conserved quantities along geodesics, there is a conformal Killing tensor, MN , for which one has: D D MN dzM dzN d d = F ( ) gMN and two quadratic \energies," one involving vr dd alone. Properties (ii) and (iii) together mean that not only is the massless wave equation separable, but its separability properties are precisely those of the at-space base metric written in spherical bipolars (2.4). In particular, the angular modes on S are elementary: they are simply the standard spherical harmonics on a round S3! Therefore the solutions of the massless wave equation have an expansion in terms of functions of r alone, multiplied by Jacobi polynomials in cos2 . Thus, most of the interesting physics is encoded in the radial equation and in the functions of r alone that de ne g^ . Finally, property (iii) suggests a rather interesting conjecture arising from the general lore of consistent truncation of supergravity theories compacti ed on spheres. Typically, purely internal, higher-dimensional excitations reduce to scalar elds in the lowerdimensional theory. Conversely, one of the most complicated aspects of obtaining \uplift" 4See also the recent review on geodesic integrability in black-hole backgrounds [36]. d dr alone and the other involving v { 6 { formulae for consistent truncations is the way in which lower-dimensional scalars encode the details of the higher-dimensional elds. For maximally supersymmetric theories on spheres, there is now a large literature on this, but one of the earliest breakthroughs were the metric uplift formulae [9, 10, 37, 38]. These formulae gave complete and explicit expressions, in terms of the lower-dimensional scalar elds, for the inverse metric projected onto the internal manifold. One of the simple consequences of this formula is that if the inverse metric retains its original round form, then it means that the lower-dimensional scalars are essentially trivial. This piece of lore suggests that in the solutions we are considering, there are no fundamental, lower-dimensional scalar excitations arising from the six-dimensional metric: all the internal physics is encoded in the vector multiplets that descend from the Bi . There are several caveats that come with this comment. First, we have not analyzed the tensor gauge elds to determine whether or not a complete, consistent truncation containing the above solutions exists. Since these tensor elds have non-trivial dependence on the S3 directions, they will descend to massive elds in three dimensions. Moreover, while important results have been obtained in [10], the general consistent truncation formulae have not been established for the S3 compacti cations considered here. In addition, the formulae that have been established in other reductions are based on massless vector elds that descend through Killing vectors on the sphere. For the tensor elds and for some of the metric components in the (2; 1; n) family, we will need to allow more general classes of elds, Bi , in which the internal components involve higher harmonics, yielding massive vector elds on K. As we shall comment on below, it is not clear how restricted an ansatz might be necessary in order to obtain a consistent truncation. It is also important to note that, from the three-dimensional perspective, Abelian vector elds can trivially be re-written as scalars. On the other hand, there are subtleties in doing this for non-Abelian elds and with o -shell supermultiplet structure (see, for example, [39]) and so the idea that there are only excitations of three-dimensional vector elds, descending from metric modes on S3, may be given some more precise formulation. The bottom line is that the supergravity lore on consistent sphere truncations suggests that property (iii) might imply that the only degrees of freedom that are being activated in our solutions are the vector elds encoded in Bi and that there are no other independent shape modes or lower-dimensional scalars coming from the six-dimensional metric. 3 Single-mode superstrata In this section we review the construction of superstrata, before focusing on the set of such solutions that involve a single-mode excitation. This will provide some background and allow us to set up notation to be used when we present our main results in the next section. 3.1 D1-D5-P superstrata The superstrata constructed to date have been obtained [1, 4, 32] by adding momentum waves to the background of the circular supertube [40{44]. The starting point is therefore { 7 { 1=2 e1 = (r2 + a2)1=2 dr ; (1) (2) (3) dr ^ d (r2 + a2) cos r dr ^ d'1 r One may then write: and introduce a standard basis for the self-dual two forms: + r sin d'1 ^ d'2 = r2 + a2 dr ^ d'2 + tan d ^ d'1 = 1 1 (r2 + a2) 21 cos 12 (r2 + a2) 21 cos (e1 ^ e2 + e3 ^ e4) ; (e1 ^ e4 + e2 ^ e3) ; (3.3) HJEP1(207) 1 1 2 r sin cot d ^ d'2 = (e1 ^ e3 e2 ^ e4) : to take the vector eld to be that of the standard magnetic ux of the supertube: = p We use the following frames on the four dimensional base, B, with metric (2.4): = Z1 (1) + Z2 (2) 2 Z4 (4) ; 1 2 DF (3.10) (4) ^ (4) : p 2 Ry a2 2 (3) = d = ((r2 + a2) cos2 (2) r2 sin2 (3)) : In particular, (3) is self-dual. The rst part of the solution is de ned by three more potential functions, ZI , and magnetic 2-forms, (I), I = 1; 2; 4, that are required to satisfy the \ rst layer" of the linear system of equations governing all supersymmetric solutions of this theory: The operator, D, acting on a p-form with legs on the four-dimensional base (and possibly depending on v), is de ned by: where d(4) denotes the exterior derivative on B. The warp factor P in (2.2) is then determined by a quadratic form in the electric potentials: The remaining metric quantities are determined by the \second layer" of BPS equations: D d(4) P = Z1 Z2 Z42 : 1 2 4 Z2) 4 ^ (1) (2) { 8 { (3.1) (3.2) (3.4) (3.5) (3.6) (3.7) (3.8) (3.9) We will study solutions to the BPS equations with mode dependence of the form: We also de ne: p 2 Ry k;m;n (m + n) v + (k m) '1 m '2 : k;m;n ak rn (r2 + a2) k+2n sink m cosm : The smoothness of the solutions requires k to be a positive integer and m, n to be nonnegative integers with m k. This restriction has a clear holographic interpretation in the description of the dual CFT states [1, 4, 32]. The most general solution to the rst layer is known for the single-bubble solutions that are built upon the circular supertube [1, 4, 45{47]. This family of solutions can be represented by a superposition of the following single-mode solutions for the pair (Z4; (4)): HJEP1(207) (3.11) (3.12) (3.13) Z4 = b4 (4) = Ry p k;m;n cos k;m;n ; 2 b4 k;m;n h (m + n) r sin + n + cos k;m;n m m k n k 1 + 1 r sin (2) + n sin k;m;n m k 1 (1) (3) i ; with similar expressions for (Z1; (2)) and (Z2; (1)), with a priori independent coe cients. The general families of regular solutions for the second layer have not been classi ed. Classes of single-mode solutions are known [1, 2, 4, 32] and some multi-mode solutions have been obtained [4]. Here we will focus entirely on the families of single-mode solutions obtained in [1] and further studied in [2]. 3.2 Coi ured single-mode solutions In the maximally-rotating supertube solution, the data of the rst layer of BPS equations takes the following simple form: Z1 = Q1 ; Z2 = Q5 ; Z4 = 0 ; (I) = 0 ; I = 1; 2; 4 : (3.14) To this solution we add a single uctuating mode by taking Z4, 4 to be given by (3.13) and by taking: Z1 = (2) = Q1 b 2 2 b4 p Ry Observe that the Fourier frequencies appearing in (Z1; (2)) are twice those appearing in (Z4; (4)) and that the Fourier coe cients of these modes have been tuned in terms of the +cos 2k;2m;2n 2m +1 { 9 { square of the Fourier coe cients of (Z4; (4)). This is an example of the procedure known as \coi uring" [48, 49]. The problem is that generic uctuations for the solutions to the rst layer of BPS equations typically lead to singular solutions in the second layer. This may be related to the non-linear instabilities that have been suggested in [11{14]; this is currently under investigation. Coi uring solves this problem by tuning other excitations to remove the singularities in the solutions to the second layer of BPS equations. For a single mode the result is particularly simple: all dependence on (v; '1; '2) cancels in the sources for the second layer of BPS equations and in the warp factor, P. As a result, for a single mode, the entire metric (2.2) is independent of (v; '1; '2). All that remains of the uctuations is the \RMS values" proportional to b24. The warp factor P now reduces to: P = Q1Q5 2 1 b 2 4 2a2 + b2 2k;2m;2n It was shown in [1] that this is positive de nite for all r and . Next, one must solve the second layer of BPS equations. Since the coi ured sources are independent of (v; '1; '2), this means that we can use the following Ansatz: (j++i1)N1 (J +1)m (L 1 J 3 1)n m! n! j00ik ; !1 d'1 + !2 d'2 = !0 + !^1(r; ) d'1 + !^2(r; ) d'2 ; F = F (r; ) ; where !0 the angular momentum vector of the round supertube: !0 Ry a2 p 2 (sin2 d'1 + cos2 d'2) ; and where !^1; !^2 and F are determined by solving (3.10), with the sources on the righthand side given by the uctuating solution to the rst layer of BPS equations described above. The general family of (k; m; n) single-mode superstratum solution was obtained in [1]. Regularity at r = 0; = =2 (the \supertube regularity" condition) imposes the constraint: Ry2 Q1Q5 = a2 + b 2 2 ; b 2 k m k + n 1 n 1 b42 : The explicit solutions for (k; m; n) = (1; 0; n) and (k; m; n) = (2; 1; n) were studied in detail in [1] and [2] respectively, and we will exhibit them momentarily. In [1] it was shown that the complete (k; m; n) solution has the following values of the conserved ve-dimensional angular momenta j, ~j, and y-momentum nP: j = N 2 a2 + k m b2 ; ~j = N a2; 2 nP = N m + n 2 b ; 2 k where N n1n5Ry2=(Q1Q5), and n1; n5 are the numbers of D1 and D5 branes. It was proposed in [1] that these solutions are holographic duals of coherent superpositions of CFT states of the form: (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) for all values of N1 such that N1 + kNk;m;n = N . For an explanation of the above notation, see [1]. The values of the conserved charges imply that, within the coherent superposition, the average numbers of j++i1 and j00ik strands are given by N a2 and N b2=(2k) respectively. 4 The special families of superstrata metrics We now examine the details of the solutions for which the parameters (k; m; n) take the values (1; 0; n), (2; 1; n), and (2; 0; n). The solution for (k = 2; m = 0) is included simply to illustrate that properties (i) and (ii) do not hold in general, since neither property holds for this solution. We will therefore not discuss the details of this particular solution beyond writing down the metric, and we will focus on the other two families. For k = 1, m = 0 and general n > 0, the solution to the second layer of BPS equations is [1]: F = b 2 a2 For k = 2, m = 1 and general n > 0, we have [2]:5 F = !1 = p !2 = p b 2 (n + 1) a2 r2n sin2 ; r 2 2 (r2 + a2)n+1 2 (n + 1) + b24 r2(n+1) cos2 2 (r2 + a2)n+2 Finally, for k = 2, m = 0 the solution to the second layer of BPS equations is: Regularity requires that b and b4 are related via the general relation (3.19), which evalub2 = b42 for k = 1 and b2 = for k = 2 : (4.4) b 2 4 2(n + 1) 5To arrive at the following expression we have taken the results from section 6.1 of [2] and undone the gauge transformations described in section 2.2 of that paper. We have thus ensured that (4.1) and (4.2) are solutions in the conventions of this paper. !2 = p Ry 2 Ry 2 ates to: Ry Ry 2 2 b 2 + a2 cos2 ; (4.2) F = (n+1)2 a4 na2 r 4 Note that killing o all the non-trivial modes by setting b4 = b = 0 in (4.1){(4.3) reduces these expressions to !0 in (3.18). The value of P in the starting supertube solution is Q1Q5= 2. It is convenient to factor this o and introduce the quantity: p P pQ1Q5 = s 1 where the functions, Fi(r), are de ned by: F0(r) F2(r) 1 1 r2n a2 b2 ; (2a2 + b2) (r2 + a2)n+1 : F1(r) r2n a 6 b2 (2a2 + b2) r2 F0(r) ; From (4.6) it is elementary to evaluate the determinant of the internal metric along S. Recalling that (2.7) de nes the warp factor required by dimensional reduction, we nd: In (4.6) we have extracted this warp factor from the rst part of the metric and so the metric terms inside the rst set of square brackets yield the metric, g^ , on K de ned in (2.6). The resulting three-dimensional metric is, indeed, purely a function of r, g^ dx dx = pQ1Q5 " F2(r) dr2 + We de ne the following one-forms on the three-dimensional base, K: (a2(du dv) b2 F0(r)dv) : 2 = F2(r) : 2 a2 r2 (r2 + a2) F2(r) F1(r) Ry2 dv + 1 du2 F1(r) p 2 (2a2 + b2) F2(r) Ry (4.6) 2 ; (4.7) (4.8) (4.9) (4.10) We now observe that the o -diagonal components, Bi of the bration form of the metric (2.6) can be written as Bi dx = K(1i) A(1) + K(2i) A(2) ; (4.11) only is the metric g^ components of the metric on S where K(1M) = (0; 0; 0; 0; 1; 0) and K(2M) = (0; 0; 0; 0; 0; 1) are the components of the Killing the vector elds, A(1) and A(2) are massless electromagnetic potentials on K. Thus, not independent of the coordinates on S3, but so are the dynamical . In light of Property (iii) and the comments made at the end of section 2 about the absence of scalar excitations and shape modes, the dynamics of the six-dimensional metric excitations in this solution reduces to dynamics of the metric and massless vector elds on K. Of course, one should recall that in the complete sixdimensional solution, the three-form elds depend upon v; '1 and '2, and so do not reduce to massless elds in three dimensions. It is, however, still possible that such tensor gauge modes give rise, in a consistent truncation, to a collection of massless and massive elds on K. This possibility is currently under investigation. 4.1.2 Since the six-dimensional metric is independent of (u; v; '1; '2), this means that the corresponding momenta are conserved: L1 = K(1)M d dzM ; L2 = K(2)M d dzM ; P = K(3)M d dzM ; E = K(4)M d dzM ; (4.12) In addition, there is the standard quadratic conserved quantity coming from the metric: " dzM dzN d d : These conservation laws determine all the velocities except vr d dr and v In principle (4.13) allows exchange of energy between vr and v , and this could generate interesting trapping of geodesics: as a particle falls in, some of its vr is traded for v and thus the particle may lose radial momentum and be prevented from returning to where it started. However, at least for null geodesics in the (1; 0; n) family of metrics, there is a hidden symmetry: there is an additional conserved quantity, that is quadratic in momenta. The additional conserved quantity can be found by separating variables in the massless Hamilton-Jacobi equation, and takes the form: One can verify that for any geodesic one has which vanishes on null geodesics. MN d d dzM dzN d d = Ry v Q1Q5 2 v2 + L2 1 sin2 + L2 cos2 dzM dzN d d ; (4.13) dd . (4.14) (4.15) F2(r) n = 1 and, as (4.17) implies, the size of the dip decreases with n. Were it not for the presence of 2 in (4.14), this conserved quantity would be the total angular momentum on the round S3, and the motion on S would be essentially decoupled from that on K. However, because of the factor of On the other hand, the factor of 2 only exerts a minor in uence on geodesic motion. To 2, these motions are not decoupled. see this, rst observe that 2jr=0 = 1 ; 2 ! 1 as r ! 1 ; 2 j = 2 = F2(r) : at In fact 2 is very close to 1 for most values of (r; ), and its maximum deviation from 1 is = 2 where it is given by F2(r). The function F2(r) is minimized at r = apn, and has a minimum value of b 2 nn 1 2a2 + b2 (n + 1)n+1 > 3 4 : Moreover, as one can see in gure 1, the variation from 1 takes place in a short interval around r = apn. The region around r = apn is also the region where the microstate structure, in the form of momentum-carrying waves, is concentrated. This means that v will increase brie y as it passes through r = apn and this can, in turn, change the value of and the value of vr. However, this e ect is quite localized and makes little di erence to the asymptotic values or vr and r. A free particle falling in from outside the throat will bounce o the center and escape the throat. The only geodesics that are a ected signi cantly by the microstate structure are the ones that are already (4.16) (4.17) localized near r = apn. 4.1.3 The wave equation Consider the six-dimensional scalar wave equation 1 p = pQ1Q5 ; (4.18) a2(! + p + q1)2 K = ( + M 2) K ; S = S ; (4.21) HJEP1(207) for some eigenvalue . Observe that the second equation (4.21) is the eigenvalue problem for the Laplace operator on the round S3. The regular modes are therefore given in terms of Jacobi S( ) = sinq1 cosq2 P (q1;q2)(cos 2 ) ; j P (q1;q2) (x) = (q1 + 1)j 2F1 j! 1 2 j; 1 + (` + q1 + q2) ; q1 + 1; (1 x) ; 1 2 m) is Pochhammer's symbol. The quantum numbers (`; j) are = `(` + 2) ; j = (` q1 q2) : For the modes (4.19) to be single-valued and regular, and for Pj(q1;q2) to be a polynomial, one must have where the factor of 1 has been included in the \mass term" on the right-hand side for reasons that will become apparent below. Consider a generic mode for of the form = K(r) S( ) ei Rpy2 ! u+ Ry2 pv+q1 '1+q2 '2 : p One then nds that the wave equation separates, yielding: (4.19) (4.20) (4.22) (4.23) (4.24) (4.25) (4.26) polynomials: where and (y)j = Qj m=0(y de ned by q1; q2; `; j 2 Z ; q1; q2 0 ; ` q1 + q2 mod 2 : The radial equation is considerably more involved, because of the presence of terms proportional to b2, which encode the massless scattering from the detailed structure of the superstratum. Note that setting r = a sinh turns the di erential operator part of (4.20) to a more canonical form: 1 sinh cosh Note also that in the limit b ! 0, the background becomes the decoupling limit of the circular supertube solution, and the radial equation simply becomes the hypergeometric equation [50]. It has been known for some time that the massless scalar wave equation is separable in the circular supertube solution [50], as well as in solutions obtained by spectral ow thereof [ 25, 51, 52 ], and related black hole solutions [53]. The (1; 0; n) family of single-mode superstrata is considerably more complicated than these solutions, and so it is quite remarkable that separability is preserved. Finally, note that the mass term in (4.18) has descended to a mass term in three dimensions, as is evident in (4.20). Indeed the total three-dimensional mass of the scalar eld is the sum of the explicit mass, M 2, and the eigenvalue, . We now analyze the (2; 1; n) family of solutions. The warp factor, , now takes the form: = s 1 Note that, compared to the corresponding warp factor in the (1; 0; n) solutions, (4.5), the warp factor involves a higher harmonic mode with a stronger fall-o at in nity. This means that the non-trivial pro le in is even smaller than the pro les depicted in gure 1. The pro le is sharply peaked around r = r n 2 a : = '1 + '2 ; = '2 '1 ; (4.28) (4.29) (4.30) 2 (4.31) (4.32) We introduce the coordinates and the functions, Hi(r): H0(r) H2(r) 1 1 r2n a4 (du + dv) + (2a2 + b2) r2 H1(r) dv In terms of these, the six-dimensional metric, (2.2), can be re-written as: dv2 4 + pQ1Q5 d 2 + H2(r) 4 + d + A^( ) 2 + H2(r) 4 cos2(2 )H2(r) + sin2(2 ) cos 2 d + A^( ) d + A^( ) d + A^( ) 2 ; where the vector elds A^( ) and A^( ) are given by: A^( ) = A^( ) = p p Ry 2 2a2du b2H0(r)dv 1 2 dv + cos 2 (2a2 + b2) 1 H2(r) H2(r) : a 2 r 2 2a2 + b2 (du + dv) + a2 H1(r)dv ; The dimensionally-reduced metric on K, de ned in (2.6), is then: dr2 + As before, it is elementary to evaluate the determinant of the internal metric along S and use (2.7) to obtain the warp factor required by dimensional reduction: a4 (du+dv)+(2a2 +b2) r2 H1(r) dv which is, again, purely a function of r. Like the (1; 0; n) vector elds in (4.10), the vector eld A^( ) is independent of the coordinates on the S3 and, when incorporated in the metric, is multiplied by a Killing vector. This means that it reduces to a massless Kaluza-Klein vector eld on K. However, A^( ), while also multiplying Killing vectors in the metric, has terms that are independent of as well as terms proportional to cos 2 . The former are constant multiples of dv and are thus pure gauge. The latter also depend on r and therefore represent non-trivial pro les for massive vector elds on K. The metric of the (2; 1; n) family thus produces both massive and massless KK vector elds on K. 4.2.1 Geodesics and separability For the (2; 1; n) family of solutions, the massless wave equation and the Hamilton-Jacobi equation for null geodesics are separable only for either vanishing frequency, or for a speci c choice of angular modes on S3. Speci cally, if one seeks modes of the form (4.19) then one nds something very similar to (4.20){(4.21) except for a single problematic term. One nds: 1 1 + 1 1 S sin cos + F (!; p; q1; q2; n; r) a2b2 (n + 1) G(!; q1; q2; n; r; ) = 0 q 2 1 sin2 + q 2 2 cos2 where F (!; p; q1; q2; n; r) is a complicated function of the coordinate r and the mode numbers !; p; q1; q2 and n. The function G(!; q1; q2; n; r; ) is given by: G(!; q1; q2; n; r; ) ! (q1 + q2) (r2 + a2)n+2 cos 2 r2n as r ! 0 and r ! 1. equation: and expresses the failure of separability. Note that this term vanishes if ! = 0 or q2 = q1. Moreover, the function G is strongly peaked at the dimple of , (4.28), and vanishes rapidly One nds a similar result upon attempting to separate the massless Hamilton-Jacobi = 0 : (4.33) (4.34) 2 ; S (4.35) (4.36) (4.37) Gb(E; `1; `2; n; r; ) E (`1 + `2) (r2 + a2)n+2 cos 2 : r2n This is manifestly the direct parallel of (4.36). Moreover, if either E vanishes or `1 + `2 vanishes, then Gb 0 and Hamilton-Jacobi theory tells us that is a conserved quantity. Note that one has 2 P = Ry2 (2a2 + b2) 1 = Ry2 (2a2 + b2) 2 ; one obtains an equation of the form: (S0( ))2 + ` 2 1 sin2 + ` 2 2 cos2 where + (r2 + a2) (K0(r))2 + Fb(E; `0; `1; `2; n; r) + (n + 1) a2b2 Gb(E; `1; `2; n; r; ) = 0 ; where is given in (4.27). Thus (4.41) is the analogue of the conserved quantity (4.14). However, (4.41) is only conserved for E = 0 or `1 = `2. Recall that the momentum modes that underlie our solution depend upon the angles according to (3.11), which now has the form: p Ry 2 2;1;n (n + 1) v + ('1 '2) : Thus the conservation and separation conditions, `1 = `2 and q2 = q1, mean that the geodesic or wave must have the same angular dependence on the S3 as the underlying momentum modes. 5 Discussion In this paper we have found that two in nite families of superstratum solutions have quite remarkable integrability properties for null geodesics. One of the families, the (1; 0; n) family, has a separable massless Klein-Gordon equation and a complete set of conserved quantities for null geodesics. The other family, the (2; 1; n) family, has a separable massless Klein-Gordon equation and a complete set of conserved quantities only for a constrained set of angular momenta on the S3. For the (2; 1; n) family (and for the (2; 0; n) family), the failure of separation and failure of conservation is sharply localized in the region of the solution where the momentum density is concentrated. We further found that the metrics of these families of solutions can be reduced to interesting sets of degrees of freedom in (2 + 1)-dimensions. The conservation laws and separability of the massless scalar wave equation for the (1; 0; n) family means that this solution is readily amenable to detailed scattering calculations. On the other hand, because of its integrability, this solution is likely to exhibit some quite atypical behavior, particularly when it comes to the spectrum. An interesting question to investigate is whether, and in what regime of parameters, a given microstate geometry can capture or trap incoming particles. It was recently argued that given any supersymmetric microstate geometry in six dimensions, there should exist a stably-trapped null geodesic passing through every point of the spacetime [ 11 ]. These null geodesics have tangent vector @=@u in our notation, so correspond to massless particles moving purely in the y direction. In the geodesic approximation, massless particles following such geodesics do not fall (deeper) into the throat. The main heuristic argument of [ 11 ] considers such a particle that is coupled to gravitational radiation and other massless elds, such that the probe gradually radiates some of its energy into these other elds, thus evolving to follow geodesics of progressively lower energy. In this way a massless particle can, slowly, descend the throat. However, within the geodesic problem, one can ask whether an infalling particle with non-zero radial momentum, falling from outside the throat, can be de ected non-trivially from the region of the metric at the bottom of the throat and, through this de ection, remain in the throat for arbitrarily long periods of time, as seen from in nity. If the \radial kinetic energy," 12 vr2, is the only kinetic energy term that appears in a particular conservation law, then, just as in any orbit problem, an infalling particle falling from outside the throat will simply rebound and escape: it will not be captured and trapped deep within the throat. For such capture to occur, particles and waves must be able to scatter \radial kinetic energy," 12 vr2, into \angular kinetic energy." A complete set of conserved quantities is thus largely antithetical to such behavior, although the conserved quantity, (4.14), depends on both r and via (4.5) and so, in principle, it is possible that changing along the trajectory can result in the loss of some radial kinetic energy. However, in practice, in our solutions the dependence dies out extremely rapidly for large r, and so changing will have only a minimal e ect upon the return of the particle to large distances. If a given solution does allow a signi cant de ection of radial momentum to angular momentum, the angular motion can potentially prevent the particle from escaping the throat for a long time. Our results imply that the (1; 0; n) family of solutions is likely the wrong place to look for such behavior. However, the conservation laws present for the (1; 0; n) family are not present for generic (k; m; n) superstrata, so such solutions should allow the scattering of vr into angular motion, and it would be interesting to investigate whether this can lead to geodesics that describe infalling particles with non-zero radial momentum, falling from outside the throat, becoming trapped deep inside the throat for long periods of time. One way to study this behavior analytically could be to use the (2; 1; n) family (for which geodesics/waves with `1+`2 = 0 or q1+q2 = 0 are integrable) and examine perturbatively how waves are scattered into angular directions for small jL = `1 + `2 or small q1 + q2. It would be particularly interesting to investigate the timescale associated with this trapping. This will depend on the depth of the throat, on how radial motion is converted into angular motion, and on whether or not the trapping is chaotic. It might be that the only way in which a particle can return to large distances after having scattered o the microstate structure once is to scatter o it again in exactly the right manner as to restore enough radial kinetic energy. This could lead to extremely large return times | a desirable feature if one is to construct microstate geometries that describe typical black hole states. Finally, the results presented here underline the fact that the (1; 0; n) and (2; 1; n) families have remarkably stringent constraints on their structure that suggests that the full six-dimensional solutions might be written in a form that would come from a consistent truncation to (2 + 1)-dimensions. In particular, the Fourier expansions of the (2 + 1)dimensional elds may only have non-trivial dependence on one variable, r. If such a structure indeed exists, it would be very interesting to investigate the existence or otherwise of a consistent truncation containing these solutions. In doing so, an interesting question will be to determine whether or not any consistent truncation ansatz will require some form of coi uring to be built in. If a consistent truncation involving the modes of the tensor gauge elds exists, then this could provide a powerful new route for the construction non-supersymmetric solutions building on [54{56]: it would reduce the non-linear, nonBPS supergravity dynamics in the system of [54, 56] from functions of two variables in six dimensions to the far more tractable problem of functions of one variable in (2 + 1)dimensions. These questions are currently under investigation. Acknowledgments NPW is very grateful to the IPhT of CEA-Saclay, and IB and DT to the Centro de Ciencias de Benasque, for hospitality during this project. We would like to thank Stefano Giusto, Mariana Gran~a, Monica Guica, Stefanos Katmadas, Emil Martinec, Rodolfo Russo, Masaki Shigemori and Charles Strickland-Constable for interesting discussions. 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Iosif Bena, David Turton, Robert Walker, Nicholas P. Warner. Integrability and black-hole microstate geometries, Journal of High Energy Physics, 2017, 21, DOI: 10.1007/JHEP11(2017)021