Momentum fractionation on superstrata

Journal of High Energy Physics, May 2016

Superstrata are bound states in string theory that carry D1, D5, and momentum charges, and whose supergravity descriptions are parameterized by arbitrary functions of (at least) two variables. In the D1-D5 CFT, typical three-charge states reside in high-degree twisted sectors, and their momentum charge is carried by modes that individually have fractional momentum. Understanding this momentum fractionation holographically is crucial for understanding typical black-hole microstates in this system. We use solution-generating techniques to add momentum to a multi-wound supertube and thereby construct the first examples of asymptotically-flat superstrata. The resulting supergravity solutions are horizonless and smooth up to well-understood orbifold singularities. Upon taking the AdS3 decoupling limit, our solutions are dual to CFT states with momentum fractionation. We give a precise proposal for these dual CFT states. Our construction establishes the very nontrivial fact that large classes of CFT states with momentum fractionation can be realized in the bulk as smooth horizonless supergravity solutions.

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Momentum fractionation on superstrata

HJE Momentum fractionation on superstrata Iosif Bena 0 1 2 4 5 Emil Martinec 0 1 2 3 5 David Turton 0 1 2 4 5 Nicholas P. Warner 0 1 2 5 0 University of Southern California , USA 1 5640 S. Ellis Ave. , Chicago, IL 60637-1433 , U.S.A 2 CEA, CNRS , F-91191 Gif sur Yvette , France 3 Enrico Fermi Institute and Department of Physics, University of Chicago 4 Institut de Physique Th ́eorique, Universit ́e Paris Saclay 5 Los Angeles , CA 90089 , U.S.A Superstrata are bound states in string theory that carry D1, D5, and momentum charges, and whose supergravity descriptions are parameterized by arbitrary functions of (at least) two variables. In the D1-D5 CFT, typical three-charge states reside in highdegree twisted sectors, and their momentum charge is carried by modes that individually have fractional momentum. Understanding this momentum fractionation holographically is crucial for understanding typical black-hole microstates in this system. We use solutiongenerating techniques to add momentum to a multi-wound supertube and thereby construct the first examples of asymptotically-flat superstrata. The resulting supergravity solutions are horizonless and smooth up to well-understood orbifold singularities. Upon taking the AdS3 decoupling limit, our solutions are dual to CFT states with momentum fractionation. We give a precise proposal for these dual CFT states. Our construction establishes the very nontrivial fact that large classes of CFT states with momentum fractionation can be realized in the bulk as smooth horizonless supergravity solutions. Black Holes in String Theory; AdS-CFT Correspondence 1 Introduction 2 BPS solutions in supergravity The BPS equations in six dimensions BPS solutions in five dimensions 2.3 A round supertube in flat space 4 Adding momentum to the supertube The first layer of equations The second layer of equations The complete angular momentum vector Coiffuring and regularity Regularity bounds and CTC’s 2.1 2.2 3.1 3.2 4.1 4.2 4.3 4.4 4.5 5.1 5.2 5.3 5.4 5.5 – 1 – 3 Supertubes with momentum via spectral interchange Spectral interchange in general Spectral interchange: an example 3.3 The Green function and mode expansions on an R4/Zk base 5 Dual CFT states 6 Discussion A The BMPV black hole B Reduction to five dimensions B.1 Reduction 1 B.2 Reduction 2 C The lowest Style 1 modes CFT spectral flow to 18 -BPS states CFT duals of our superstrata Comparison of conserved charges 5.6 Comments on momentum fractionation 41 -BPS operators: linearized supergravity modes 41 -BPS states: twisted sector ground states of the CFT Introduction String theory has been successful in counting the microstates of black holes in the regime of parameters where stringy effects overwhelm gravitational effects at the horizon scale. When supersymmetry is present, this counting carries over to the regime of parameters where gravitational effects are dominant at the horizon scale, and the entropy of these microstates reproduces the Bekenstein-Hawking entropy of the black hole [1, 2]. However, the exploration of the implications of this achievement for resolving the information paradox [3] and for understanding the physics of an infalling observer [4–7] is still in its infancy. Indeed, very little is known about the fate of the individual stringy microstates, counted in the zero-gravity regime, as one increases the gravitational coupling and goes to the regime in which the configuration corresponds to a classical black hole with a large event horizon. There are several possibilities as to what this fate might be. One is that, as gravity becomes stronger, all these microstates develop a horizon and end up looking identical to the black hole [8–10]. Another is that some of the microstates that one constructs at zero gravitational coupling will develop a horizon, and others will remain horizonless. A third possibility is that none of these microstates develop a horizon, and they all grow into horizon-sized bound states that have the same mass, charges and angular momentum as the black hole, but have no horizon [11–18]. There are then a range of “sub-possibilities”: at one extreme, typical black-hole microstates would not be describable in supergravity, but will be intrinsically quantum or non-geometrical; at the other extreme, in the sector dual to the typical microstates, one could find a basis of Hilbert space vectors that correspond to coherent states that have a supergravity description, or at least a stringy limit thereof. In the context of the AdS-CFT correspondence [19], one can similarly ask whether a typical CFT microstate corresponds to a classical black hole with an event horizon, or to some horizonless configuration. The latter might either be impossible to describe in supergravity because of large quantum fluctuations or stringy corrections, or might be described using a Hilbert state basis given by smooth low-curvature solutions, or might correspond to some hybrid configuration (such as an intrinsically quantum configuration lying in a smooth, horizonless supergravity solution). There exist pieces of evidence that can be taken as bringing support to any of these possible outcomes, some founded on calculations, and some based more on intuition and conjecture. Perhaps the strongest evidence that at least some microstates become smooth horizonless supergravity solutions at strong gravitational coupling comes from the explicit construction of numerous families of smooth horizonless solutions that have the same charges as black holes [20]. The largest family of solutions are parametrized by arbitrary continuous functions of two variables [21], and come from the back-reaction of certain families of superstrata [22]. Superstrata are string theory bound states whose counting has been argued to reproduce a finite fraction of the entropy of three-charge supersymmetric black holes [23]. However, even if the existence of these large families of solutions rules out the possibility that all the microstates one counts at zero gravity develop a horizon, it does not prove that all microstates remain horizonless, nor does it establish whether typical horizonless – 2 – configurations are smooth and describable in supergravity, or are instead non-geometric or strongly-curved. For example, it has been argued [24] that for the two-charge D1-D5 black hole, the typical states of the dual symmetric product orbifold CFT [ 19, 25–28 ] are not well-described by the microstate geometries of [11, 29, 30] when the average harmonic of the two-charge profile function becomes larger than √N1N5. The harmonics of the profile function correspond to the winding of strands in the D1-D5 orbifold CFT; since typical three-charge microstates come from adding momentum to CFT strands whose length is of order N1N5, one might naively conclude that all typical three-charge microstate geometries would be strongly-curved and hence not describable by supergravity. There are also arguments that the bulk configurations dual to typical CFT states will be non-geometrical. One such argument comes from an analysis of the possible supertube transitions that can occur in three-charge systems, which indicate that the configurations resulting from these transitions will be generically non-geometric [31, 32]. It has also been suggested that the states that carry fractionated momentum modes, which are the typical states that contribute in the entropy counting, will involve multi-valued wavefunctions [21]. Furthermore, there are also conjectures that when tracking microstates of the D1-D5-P system to the regime of parameters where gravity becomes important, only very few states will give rise to horizonless geometries, while most states will correspond to a black hole with a horizon [33]. According to this perspective, the more typical the state, the larger the likelihood that its bulk dual will not be a horizonless solution, but will be a solution with a horizon. The purpose of this paper is to provide evidence that these alternative scenarios are not realized, by showing that highly-nontrivial CFT states whose momentum is carried by fractionated carriers are dual to smooth horizonless supergravity solutions (with localized orbifold singularities). We construct these solutions using a combination of two solutiongenerating techniques: Spectral interchange (also known as spectral inversion) and adding charge density oscillations to a supertube. Spectral interchange is a transformation of the D1-D5(-P) BPS solutions that interchanges the null coordinate along the D1 and D5 branes, v = t + y, with the Gibbons-Hawking fiber of the transverse space [34, 35]. Modifying the charge density distribution along the supertube source profile has been studied, for example, in [29, 36, 37]. In this paper we show that by combining these techniques one can add y-momentum to a seed solution with D1 and D5 charges, as follows: first perform a spectral inversion, then use a charge density oscillation to introduce ψ-dependence and associated angular momentum, then spectrally invert back to the original frame to obtain a new solution carrying v-dependence and momentum. The ψ-dependent solutions in the spectrally inverted frame can be generated by integrating the Green functions against the modified charge and angular momentum densities along the supertube. For our explicit construction, we apply this combination of techniques to a simple seed solution — a multiwound circular D1-D5 supertube. The multiwinding of the seed solution is what will allow us to study the physics of momentum fractionation. While in principle the Green function/spectral interchange method can be used to construct new general – 3 – classes of superstrata, a particular class of examples is amenable to a direct analysis of the equations governing all supersymmetric solutions of six-dimensional supergravity [38–41]. These equations determine the various potentials that enter in the supergravity solution, and are arranged in stages or layers, where the potentials to be solved for one layer satisfy linear equations sourced by the potentials determined in previous layers [40]. Our solutions are regular up to the usual orbifold singularity at the location of the multiwound supertube. We arrange the regularity of our supergravity solutions by imposing constraints on Fourier modes and coefficients; this procedure is known as coiffuring [42–44]. We find two classes of regular solutions, corresponding to two “Styles” of coiffuring. We analyze the conserved charges and other properties of the solutions. Our construction also yields the first examples of asymptotically-flat superstrata; in a particular limit our solutions contain the generalization to asymptotically-flat space of one class of the asymptotically-AdS superstrata constructed in [21]. Upon taking the decoupling limit, we obtain solutions that are asymptotically AdS3 × S3 × M (where M is either T4 or K3) and we investigate the corresponding dual CFT description. We do this by assembling a variety of clues. We observe that the relation between the y-momentum and the angular momenta of the solutions suggest that the dual CFT states involve repeated applications of fractionally moded SU(2) R-symmetry generators, and also that they can be generated by fractional spectral flow [45, 46] applied to a subset of strands of certain two-charge seed states. We then study the vevs of 41 -BPS operators and find that they have the right properties to reproduce the vevs of the supergravity fields at the linearized level, using the technology of [30, 47–49]. We find however that the supergravity regularity constraints are not visible at this order. Finally, by analyzing the possible two-charge seed solutions, we determine the precise proposal for the CFT states dual to both styles of coiffuring in supergravity. Prior to the present work, there were only two classes of supergravity solutions, one BPS and one non-BPS [50–52], which had been shown to be dual to CFT states involving momentum fractionation [53, 54].1 These states came from fractional spectral flow applied to all strands of certain two-charge states, and hence are very special. One way to see this is that the AdS region of their dual bulk solutions can be obtained from global AdS3 ×S3 by a coordinate transformation.2 In contrast, our technology produces supergravity solutions that are much more general, and cannot be written in this way. The remainder of this paper is structured as follows. In section 2, we review the class of five- and six-dimensional supergravity solutions of interest, the BPS equations they satisfy, and the multiwound circular D1-D5 supertube. In section 3, we apply the sequence of solution-generating techniques to add momentum to the seed solution. We perform a direct analysis of the BPS equations in section 4, and find two classes of regular 1There is a sense in which states obtained by the action of integer-moded generators acting on multiwound strands can be argued to involve momentum fractionation, however this fractionation is somewhat trivial and does not correspond to degrees of freedom deep inside a throat [55–57]. Thus, by “CFT states involving momentum fractionation” we mean states which cannot be written in terms of integer-moded generators acting on R-R ground states. 2 The same is true of the three-charge solutions obtained by integer spectral flow [58–60]. – 4 – HJEP05(216)4 solutions via coiffuring. In section 5, we first review the 14 -BPS states in the CFT, the 14 -BPS operators that are dual to linearized supergravity field modes, and spectral flow. We then develop the precise proposal for the CFT states dual to our supergravity solutions. Section 6 summarizes our results and discusses open questions. 2 BPS solutions in supergravity We work in type IIB string theory on R4,1 × S1 × M where M is T4 or K3. We take the size of M to be microscopic and the S1 to be macroscopic. The S1 is parameterized by the coordinate y which we take to have radius Ry, interest is an N = 1 supergravity coupled to two (anti-self-dual) tensor multiplets. This is the theory in which the first superstrata were constructed [21]; the theory contains all the fields expected from D1-D5-P string emission calculations [61]. The BPS system of equations describing all 1/8-BPS D1-D5-P solutions of this theory has been found in [41], and is a generalization of the system discussed in [38, 39] and greatly simplified in [40]. 2.1 The BPS equations in six dimensions To exploit the structure of the six-dimensional BPS equations, we work with null coordinates u and v, defined by: 1 2 u ≡ √ (t − y) , For supersymmetric solutions, the metric is required to have the local form: ds62 = − √ 2 P (dv + β) du + ω + 1 2 F (dv + β) + √ Note that we can always shift F by a constant, c, by sending u → u− 21 cv and ω → ω − 21 cβ. Given our choice of t and y coordinates in (2.2), to obtain our desired asymptotics we require that F vanishes at infinity throughout this paper. Introducing the quantities Z3 and k via3 3Note that in our conventions F is always negative. – 5 – (2.1) (2.2) (2.3) (2.4) (2.5) one can write the metric in the form 1 particular, if there are closed curves whose length in the metric ds24(B) vanishes, then it is essential that the remaining part of the metric does not make these curves time-like. The relevant condition is manifest from (2.6): the danger arises if one chooses a curve along which dy is related to the other angles such that the second square vanishes.4 We thus require that for any such curve, in the limit where the length of the curve in ds24(B) tends to zero, the one-form k acting on the tangent vector to the curve must also tend to zero (appropriately quickly). The four-dimensional base, B , has a metric, ds24, and is required to be an “almost hyper-Ka¨hler” manifold [38]. However we are going to simplify things by assuming that the base has a Gibbons-Hawking metric: where the periodicity of ψ will be given below in (2.40) and where, on the flat R3 defined by the coordinates ~y, one has: We take V to have the form 2 ∇ V = 0 , ∇~ × A~ = ∇~V . V = h + X N qj j=1 |~y − ~y(j)| , for some fixed points, ~y(j) ∈ R3, some charges, qj ∈ Z, and some constant h. We will also require that the one-form, β, is v-independent and then the BPS equations require that β has self-dual field strength: where ∗4 denotes the four-dimensional Hodge dual in the Gibbons-Hawking metric. We will also assume that β is ψ-independent and solve the self-duality by taking Θ3 ≡ dβ = ∗4dβ , β = K3 V (dψ + A) + ~σ(3) · d~y , ∇~ × ~σ(3) = − ∇~K3 . where K3 is harmonic on R3 and 4To see this, let us suppose that such curves are timelike, and let C1 be such a curve. C1 itself is not necessarily closed; denote the y values at the start and end of the curve by y1 and y2. If y2 is not equal to y1 (modulo 2πRy), consider y2 as the starting point of a new curve C2, similarly defined so that dy is related to the other angles such that the second square vanishes. By iterating, one obtains a sequence of and two scalars (one in each tensor multiplet). The scalars may be thought of as the dilaton, Φ, and axion, C0, of the IIB theory. The tensor fields of BPS solutions may be described in terms of three potential functions, Z1, Z2, Z4 and three sets of two-forms, Θ1, The BPS condition then requires a suitable generalization of the “floating brane Ansatz” [63] in which the metric warp factor and scalars are expressed in terms of the P = Z1 Z2 − Z42 , e2Φ = Z2 P 1 , C0 = Z4 Z1 . Since we are allowing the scalars and tensor gauge fields (but not β or ds24) to depend upon v, the BPS equations impose the following linear differential equations on the potentials and the two-forms (ZI , ΘI ):5 ∗4 DZ˙1 = DΘ2 , ∗4 DZ˙2 = DΘ1 , ∗4 DZ˙4 = DΘ4 , D ∗4 DZ1 = −Θ2 ∧ dβ , Θ2 = ∗4Θ2 , D ∗4 DZ2 = −Θ1 ∧ dβ , D ∗4 DZ4 = −Θ4 ∧ dβ , Θ1 = ∗4Θ1 , Θ4 = ∗4Θ4 . where the dot denotes ∂∂v , D is defined by D ≡ d˜− β ∧ ∂v , ∂ and d˜ denotes the exterior differential on the spatial base B. In (2.14)–(2.16), 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 equation. The self-duality condition reduces each ΘI to three independent components and adding in the corresponding ZJ yields four independent functional components upon which there are four constraints. If we separate the ZI into their v-independent (zero-mode) and v-dependent parts, ZI = Z(0) + Z(v), then the v-dependent parts ZI(v) satisfy simpler equations, as follows. It I I is convenient to define two-forms ξI via: ΘI ≡ ∂vξI , I = 1, 2, 4 . Then for the v-dependent parts, one can simplify the BPS equations (2.14)–(2.16) by integrating, as follows: ∗4 DZ1(v) = Dξ2 , ∗4 DZ2(v) = Dξ1 , ∗4 DZ4(v) = Dξ4 . 5We define the d-dimensional Hodge star ∗d acting on a p-form to be ∗d (dxm1 ∧ · · · ∧ dxmp ) = 1 (d − p)! dxn1 ∧ · · · ∧ dxnd−p ǫn1...nd−p m1...mp , where we use the orientation ǫ+−1234 ≡ ǫvu1234 = ǫ1234 = 1. These are the conventions used in [38] and note that they differ from the typical conventions for the Hodge dual. – 7 – (2.13) The final set of BPS equations are linear differential equations for ω and F : Dω + ∗4Dω = Z1Θ1 + Z2Θ2 − F Θ3 − 2 Z4Θ4 , (2.20) and a second-order constraint that follows from the vv component of Einstein’s equations,6 ∗4D ∗4 ω˙ + 12 DF = Z˙1Z˙2 + Z1Z¨2 + Z2Z¨1 − (Z˙4)2 − 2Z4Z¨4 − 2 ∗4 Θ1 ∧ Θ2 − Θ4 ∧ Θ4 1 = ∂v2(Z1Z2 − Z42) − (Z˙1Z˙2 − (Z˙4)2) − 2 ∗4 Θ1 ∧ Θ2 − Θ4 ∧ Θ4 . 1 closely follow that of [37]. We will assume that the magnetic 2-forms, Θ(I), are independent of the GH fiber coordinate, ψ. This means that one may use the same class of solutions as in (2.11) by introducing more harmonic functions, KI , on R3 and taking (2.21) HJEP05(216)4 with Θ(I) = dB(I) , B(I) = V −1KI (dψ + A) + ~σ(I) · d~y , ∇~ × ~σ(I) ≡ − ∇~KI . The sources in BPS equations for ZI (I = 1, 2, 3, 4) are independent of v and ψ and so the inhomogeneous solutions for the functions ZI follow the standard form: ZI = 21 CIJK V −1KJ KK + LI , where CIJK are the usual (completely symmetric) structure constants for supergravity coupled to vector multiplets. The particular theory that we use can be written in this form if one sends Z4 → −Z4 and takes ∇(24)LI = 0 . ω = µ (dψ + A) + ̟~ · d~y . D~ ≡ ∇~ − A~ ∂ψ , – 8 – (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) (2.28) with other (non-cyclically related) components equal to zero. The functions LI in (2.24) are required to be harmonic on the GH base, B, and can be allowed to depend upon all the coordinates, including ψ. Thus we have One can then make a simple Ansatz for the angular momentum, one-form ω: If one introduces the covariant derivative 6This simplified form is equivalent to (2.9b) of [64]. then the last BPS equation can be written as: (µ D~V − V D~µ ) + D~ × ̟~ + V ∂ψ ̟~ = −V ZI ∇~ V −1KI . 3 X I=1 The BPS equations have a gauge invariance: ω → ω + df and this reduces to: µ → µ + ∂ψf , ̟~ → ̟~ + D~f , V 2 ∂ψµ + D~ · ̟~ = 0 , 3 X I=1 V 2 ∇(24)µ = D~ · V ZI D~ V −1KI . ∇(24)F = V −1 V 2 ∂ψ2F + D~ · D~F . and we will impose this gauge choice. one obtains: Taking the covariant divergence, using D~, of (2.29) and using the Lorentz gauge choice, It is useful to note that the four-dimensional Laplacian may be written as: The Lorentz gauge-fixing condition, d ⋆4 ω = 0, reduces to HJEP05(216)4 The equation for µ is solved by taking: µ = 61 V −2CIJK KI KJ KK + 12 V −1KI LI + M , where, once again, M is a harmonic function on B. obtain: Finally, we can use this solution back in (2.29) to simplify the right-hand side and D~ × ̟~ + V ∂ψ ̟~ = V D~M − M D~V + KI D~LI − LI D~KI . 1 2 Once again one sees the emergence of the familiar symplectic form on the right-hand side of this equation. One can also verify that the covariant divergence (using D~) generates an identity that is trivially satisfied as a consequence of ∇~V = ∇~ × A~, (2.31), (2.34) and ∇(24)LI = ∇(24)M = 0 . An explicit, closed form for all the components of ̟~ was not given in [37], but for our solutions we will be able to construct them. 2.3 A round supertube in flat space The simplest supertube Ansatz is to take the base, B, to be flat R4 and set Θ3 and β to be that of a simple magnetic monopole. There are two convenient ways to formulate this. First, one can take β given by (2.11) and write R4 in Gibbons-Hawking form using spherical polar coordinates (ρ−, ϑ−, φ): ds42 = V −1 (dψ + A)2 + V (dρ2− + ρ2− dϑ2− + ρ2− sin2 ϑ− dφ2) , (2.37) – 9 – (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) (2.35) (2.36) where in terms of the three-dimensional Cartesian coordinates y1, y2, y3 we have where the dipole moment k is an integer. One then has: V = 1 ρ − , A = K3 = (y3 + 21 ℓ) ρ − kR ρ+ , dφ , ρ ± ≡ qy12 + y22 + (y3 ∓ 21 ℓ)2 , σ = −kR The periodicity identifications on ψ and φ are as usual subsequently want to make heavy use of the results and formalism employed in [21] and so we will use this as an opportunity to introduce the geometry and flux components that make up the second convenient description of supertubes. One starts by describing the base manifold in terms of spherical bipolar coordinates, defined by7 4 ρ+ = Σ ≡ (r2 + a2 cos2 θ) , 4 ρ− = Λ ≡ (r2 + a2 sin2 θ) , cos − = ϑ 2 (r2 + a2)1/2 Λ1/2 ψ = ϕ1 + ϕ2 , sin θ , φ = ϕ1 − ϕ2 , sin − = ϑ 2 and we choose the natural system of frames + dθ2 + (r2 + a2) sin2 θ dϕ12 + r2 cos2 θ dϕ22 , Σ1/2 r Ω(1) Ω(2) Ω(3) ≡ dr ∧ dϕ1 r and note that e1 = (r2 +a2)1/2 dr , e2 = Σ1/2 dθ , e3 = (r2 +a2)1/2 sin θ dϕ1 , e4 = r cos θ dϕ2 . (2.45) Following [21], it is convenient to introduce the self-dual two-forms Ω(1), Ω(2) and Ω(3): dr ∧ dθ ≡ (r2 + a2) cos θ + r sin θ ≡ r2 + a2 dr ∧ dϕ2 + tan θ dθ ∧ dϕ1 = 1 1 Σ (r2 + a2) 12 cos θ Σ 12 (r2 + a2) 12 cos θ (e1 ∧ e2 + e3 ∧ e4) , (e1 ∧ e4 + e2 ∧ e3) , (2.46) 1 1 Σ 2 r sin θ − cot θ dθ ∧ dϕ2 = (e1 ∧ e3 − e2 ∧ e4) , ∗4(Ω(1) ∧ Ω(1)) = ∗4(Ω(3) ∧ Ω(3)) = 2 (r2 + a2)Σ2 cos2 θ , 2 r2Σ sin2 θ , ∗4(Ω(2) ∧ Ω(2)) = 2 (r2 + a2)Σ cos2 θ , Ω(i) ∧ Ω(j) = 0, i 6= j. (2.47) 7Our spherical bipolar angles ϕ1 and ϕ2 are related to those of [21] by ϕ1here = φthere, ϕ2here = ψthere. (2.38) (2.39) (2.41) (2.42) (2.43) (2.44) The vector field β corresponding to the harmonic functions in (2.38) is βˆ = 2 kRa2 Σ (sin2 θ dϕ1 − cos2 θ dϕ2) + kR (dϕ1 + dϕ2) . To obtain flat asymptotics, we see from (2.6) that β and ω must vanish at infinity. We thus make a coordinate transformation to gauge away the constant part of βˆ, obtaining Finally there is the regularity of the metric near the supertube, which means that as one approaches Σ = 0, or r = 0, θ = π2 , the metric must remain smooth. One can easily check that the only potentially singular parts of the metric are the dϕ21 terms and these are proportional to: The vanishing of the singularity at Σ = 0 requires The two-form Θ3 = dβ is given by where c1 and c2 are constants to be determined via regularity and asymptotics. The constants Q1, Q2 and J are harmonic sources that encode charges and angular momentum. At the center of space (r = 0, θ = 0) the size of the ϕ1-circle and of the ϕ2-circle collapse to zero size as measured in the spatial base metric, ds24, in (2.4). Moreover, P goes to a constant at the center of space. It is evident from this and the discussion around (2.6) that to avoid closed time-like curves at the center of space one must have ω + β = 0 at r = 0, θ = 0. This implies: c1 = − a2 , J c2 = 2kR . In addition, ω must also fall off when r → ∞ and hence we require 2 kRa2 Σ Q1Q2 k2Ry2 . (2.48) (2.49) (2.50) (2.52) (2.53) (2.54) (2.55) (2.56) Thus supertube regularity determines the radius, a, and the angular momentum, J , in terms of the charges Q1, Q2 and the dipole charge k. We thus recover the supertube metric of [65, 66] and its Gibbons-Hawking description [67]. Having made these choices, the ψ-fiber limits to a fixed size as one approaches the supertube while the remaining part of the spatial metric limits to (in spherical polar coor(dρ2++ρ2+ dϑ2++ρ2+ sin2 ϑ+ dφ2) . (2.57) 4 ℓ √ dinates (ρ+, ϑ+, φ) centered around the supertube): 2 des4 = √Q1Q2 Setting ρ+ = 41 r+2 and using (2.43) and (2.56) one obtains: 2 des4 = √Q1Q2 hdr+2 + 14 r+2 dϑ2+ + sin2 ϑ+ dφ2 + k12 h √21 R √ which means that the metric in (2.58) represents the standard Zk orbifold of R4. Since Ry = 2 2R, one has y ∼ y + 4π 2R and so the coordinate √2 R has period 4π, y 3 Supertubes with momentum via spectral interchange The original D1-D5 supertube solution [11, 29] was defined in terms of an arbitrary profile function, F~ (vˆ), in R4. While this manifestly describes the shape of the supertube, the supertube solution is not invariant under reparameterizations of vˆ, indeed, reparameterizations encode the choice of the charge-density functions. Put differently, the supertube can be given two charge densities, ̺1 and ̺2, and an angular momentum density, ̺ˆ. However, supertube regularity and the absence of closed time-like curves (CTC’s) places two functional constraints (local analogues of (2.56)) on these densities leaving a free choice of one function. This function encodes the degrees of freedom represented by the choice of reparameterization in the original formulation. Spectral interchange can then be combined with the addition of such a charge-density fluctuation so as to generate a third (momentum) charge. 3.1 Spectral interchange in general The idea behind spectral interchange is extremely simple. When the base space, B, has a Gibbons-Hawking form then the entire solution can be written as a torus fibration over a flat R3. The torus is, of course, described by (v, ψ) and one can act on this torus with elements of GL(2, Z).8 Since these transformations are generated by simple changes of coordinate, they must map BPS solutions to BPS solutions. Some elements of this transformation group generate what are known as gauge transformations [68] and generalized spectral flows [34], that mix K3 and V . Of relevance later will be the gauge transformations: dy + √12 (σ − ̟) i2 i . (2.58) 8Technically, one should restrict to the global diffeomorphisms, SL(2, Z), but if one allows orbifolds it is sometimes convenient to use GL(2, Z). Such transformations are pure gauge in that, while they reshuffle the potentials, while leaving the physical properties of the solution invariant. Spectral interchange is a subset of the generalized spectral flow transformations [34], and is simply the modular inversion that interchanges v and ψ on the torus [35]. It corresponds to a global diffeomorphism on the fibers: v → −ψ , ψ → −v ; ⇔ V ↔ K3 , A → −ξ , ξ → −A . (3.2) This mapping also interchanges all the harmonic functions that make up the BPS solutions outlined in the previous section, as we now describe. To make the mapping more precise, we must normalize the torus angles that we interchange. The periodicity of the y circle (2.1) induces an identification on u and v. As described in (2.3) above, we parameterize this as (u, v) ∼ (u, v) + (−4πR, 4πR) . Recalling the periodicity identifications on ψ and φ given in (2.40), we see that the relevant lengths are 4πR for v and 4π for ψ. Thus the spectral interchange is more precisely written as: v R → −ψ , ψ → v − R Setting Z4 = 0 and Θ(4) = 0, spectral interchange implies that the following must also (3.3) (3.4) give a BPS solution: Ve = K3 R , Le1 = RK2 , e K3 = R V , Le2 = R K1 , e , where any ψ-dependence is converted to v-dependence in accordance with (3.4). Observe, in particular, that if the LI have some non-trivial ψ-dependence, then Ke 1, Ke 2 and Le3 and hence Fe inherit a non-trivial v-dependence. Thus the new solution involves a momentum wave and carries a momentum charge. We now implement this general idea in a specific explicit construction. 3.2 Spectral interchange: an example Our goal it to obtain a supertube with a magnetic dipole, k, and generic momentum densities and we will do this via spectral interchange. Performing spectral interchange on the round k-wound supertube (2.51), combined with a gauge transformation with parameters α 1 = − kR ¯ ¯ Qi ≡ 4 Qi , i = 1, 2 , (3.6) new BPS solution: V = K3 = 1 ρ − kR ρ+ j − R , ~y − λ1 − R , ~y − λ2 − R , ~y + v λ2 − R , ~y − ρ 1 − , ¯ kR L2 = 1 + v λ1 − R , ~y − ρ 1 − ¯ Q2 ρ+ , 1 ρ − v (3.7) (3.8) (3.9) , (3.11) (3.10) (3.12) , (3.14) (3.15) (3.16) (3.17) studied in [37], V = L1 = k ρ+ ¯ k 1 , Q1 λ1(ψ, ~y) , M = − 2 R + 2 k2R 1 Q¯1Q¯2 j(ψ, ~y) , K1 = K2 = L2 = ¯ k Q2 λ2(ψ, ~y) , 1 R , K3 = R ρ − , L3 = k + Q¯1 + Q¯2 kR2 where the λA and j are harmonic functions on R4 written as a Gibbons-Hawking space, and are sourced by normalized densities ̺1, ̺2, and ̺ˆ localized at the supertube location ~y = ~y−, that is ρ− = 0 or (y1 = 0, y2 = 0, y3 = − 2ℓ ): λA(ψ, ~y) = 4π j(ψ, ~y) = 4π Z d3y′ Z d3y′ Z 4π Z 4π 0 0 dψ′ Gb(ψ, ~y; ψ′, ~y ′) ̺A(ψ′ − kφ′)δ3(~y ′ − ~y−) , dψ′ Gb(ψ, ~y; ψ′, ~y ′) ̺ˆ(ψ′ − kφ′)δ3(~y ′ − ~y−) . (3.13) The dependence of the densities on the combination of angles ψ − kφ will become clear when we use the Green function on R4/Zk in the next subsection to construct explicit solutions. For now, we keep the discussion general to explain our overall strategy. We now transform back to the original supertube frame, first performing the inverse gauge transformation to (3.6) and then performing spectral inversion. This results in the results in a solution specified by the harmonic functions V = L1 = M = − 2 R + k ρ+ , Q¯1 1 k ρ − 1 , . , , This solution describes a supertube that is singly-wound, in a base space which is R4/Zk. The spectral interchange has thus had the effect of exchanging the original Z k orbifold at the location of the supertube for a Z k orbifold at the center of space, and the original smooth center of space has become the location of a singly-wound supertube. On this supertube in the spectrally-inverted frame, we introduce charge densities as v The form of V means that the base, B, has returned to flat R4. There is a supertube with a dipole charge k (corresponding to a pole in K3), and charges Q¯A located at ρ+ = 0. In addition, the harmonic functions λA and j describe a momentum wave along the v direction that is sourced at ρ − = 0. We have therefore succeeded in adding momentum to a standard two charge supertube solution. Spectral interchange is simply a global diffeomorphism and so regularity conditions can be imposed on the supertube in the spectral-inverted frame. Before we do this, one should note that in the original seed solution (3.10)–(3.12), the parameters, Q¯A, could be absorbed into the normalization of the charge densities, ̺A and ̺ˆ. We are therefore free to adjust them in some convenient manner and we choose to impose the constraint: As we will see, this choice will mean that one of the supertube regularity conditions is automatically satisfied for ̺A = ̺ˆ = 1. Supertube regularity with varying charge density was extensively studied in [37] (following [36]) where it was shown that the supertube (3.10)–(3.12) is regular if one imposes the following functional constraints at each point of the GH fiber: lim ρ ρ−→0 − V µ − Z3K3 lim ρ ρ−→0 − 2 V Z1Z2 − Z3(K3)2 = 0 , = 0 . Using (3.18), the first equation can be reduced to 1 kR kR (̺ˆ − 1) + Q¯1 (̺1 − 1) + Q¯2 (̺2 − 1) = 0 , where ̺A and ̺ˆ are defined in (3.13). The second regularity condition (3.20), when combined with (3.21) reduces to a simple, local constraint on the charge densities [37], ̺ˆ = ̺1 ̺2 . The regularity conditions (3.21) and (3.22) can be thought of as “coiffuring” the charge densities so as to achieve regularity. One should note that while one can certainly satisfy (3.21) using finite sets of Fourier modes, the charge density condition, (3.22), generically requires one of the Fourier series to be infinite. As we will see below, coiffuring and the holographic interpretation of the modes is somewhat simpler if one switches on (Z4, Θ4). One could repeat the foregoing analysis by introducing an additional charge density ̺4, however for ease of presentation we will continue without introducing ̺4 explicitly, and introduce (Z4, Θ4) in section 4. 3.3 The Green function and mode expansions on an R4/Zk base To construct explicit solutions of the form (3.10)–(3.12), we need the scalar Green function for a GH base space with V = ρk+ and with a source located at ρ− = 0. It is straightforward (3.18) (3.19) (3.20) (3.21) (3.22) This should not be surprising because the GH fiber is defined by (dψ + A) and, at ρ this becomes (dψ − kdφ). Thus the charge density functions and solutions will depend upon precisely this mixture of angles, explaining the form of eq. (3.13). If one expands the charge densities into Fourier modes, ̺A(ψ − kφ) = X bA,n ei n2 (ψ−kφ) , n then the solutions are elementary to obtain from the Green function using contour integration (see for example [43]): λA(ψ, ~y) = X bA,n " ρ+ − ρ− + ℓ 2 e 2i (ψ−kφ) k n ρ − ρ+ + ρ− + ℓ # n ≡ X bA,n Fˆn ρ − where Fˆn is defined through the above equation. Similarly, for j we have ̺ˆ(ψ − kφ) = X ˆbn ei n2 (ψ−kφ) , n j(ψ, ρ~−) = X bn Fˆn . n − Note that in the limit ρ− → 0 these reduce to the following simple forms: λA(ψ, ~y) → n ρ − X bA,n ei n2 (ψ−kφ) = ̺A(ψ − kφ) ρ − , j(ψ, ~y) → ̺ˆ(ψ − kφ) ρ − . Introducing spherical polar coordinates (ρ, ϑ, φ) centered at the origin (halfway between the supertube and the GH center), we observe that for ρ ≫ ℓ, to obtain this via a coordinate transformation of the standard flat R4 Green function, or one can use the general result of Page [69]. One finds that the Green function for the response at the point (ψ, ~y) caused by a source at the point (ψ′, ~y ′ = ~y−) defined by ρ− = 0 is: Gb(ψ, ~y; ψ′, ~y ′) = 1 sinhh k2 log ρρ++++ℓℓ+−ρρ−− i 16π2ρ− coshh k2 log ρρ++++ℓℓ+−ρρ−− i − cos 12 (ψ − ψ′) − k2 (φ − φ′) . (3.23) Note that this function depends upon the combination of angular coordinates: ψ − kφ . This means that Fˆn falls off as ρ− k2n at large ρ: and so higher orbifolds lead to more rapid fall-off at infinity. ρ ± ≃ ρ 1 ∓ 2 ρ cos ϑ . ℓ Fˆn ∼ 2 ρ kn ℓ(1 − cos ϑ) 2 e i2n (ψ−kφ) (3.24) − = 0, (3.25) (3.26) (3.27) (3.28) (3.29) (3.30) non-zero (Z4, Θ4). In principle one could repeat the above analysis with an additional density profile function ̺4 and analyze the modified supertube regularity conditions in the spectral inverted frame. Rather than pursue this route, we will find it more convenient to perform a direct analysis of the BPS equations using the techniques of [21] to construct our explicit solutions. This will lead to the complete solution in a manner that is well-adapted to coiffuring and holography. 4 Adding momentum to the supertube As we have seen, adding momentum to a supertube naturally leads us to consider vdependent fluctuations. We now do this by generalizing the circular supertube seed solution described in section 2.3. In this way we will also obtain the complete solution including all components of the angular-momentum vector. A natural way to construct v-dependent solutions is to introduce fluctuating chargedensity sources along the v-fiber above the center of space, r = 0, θ = 0 or ρ described in [35]. Indeed, the ψ-fiber pinches off at the center of space while the v-fiber − = 0, as remains finite: (dv + β) → (dv − 2kR dϕ2) . This means that a single-valued source introduced along the v-fiber must have a Fourier expansion with the following dependence: (4.1) (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) We will therefore seek solutions based upon these Fourier modes. Thus we define the phase: 4.1 The first layer of equations Based upon the form of eqs. (3.17), (3.26) and (3.27) and the results of [21, 35, 64, 70] it is not hard to infer a solution to the first layer of BPS equations. Define e−ip( 2vR −kϕ2) , p ∈ Z . ζ = v 2R − kϕ2 . Δ ≡ a cos θ (r2 + a2) 12 , then a somewhat lengthy computation shows that the following fields satisfy the first layer of equations (2.14)–(2.16) for some complex Fourier coefficients, b1 and b2: QA Σ ZA = 1 + Θ1 = − 2 R Θ2 = − 2 R 1 + Δkn (bA e−inζ + ¯bA einζ ) , A = 1, 2 , n Q2 Δkn h b2 e−inζ (Ω(2) + ir sin θ Ω(1)) + ¯b2 einζ (Ω(2) n Q1 Δkn h b1 e−inζ (Ω(2) + ir sin θ Ω(1)) + ¯b1 einζ (Ω(2) − ir sin θ Ω(1))i , − ir sin θ Ω(1))i , To these fields one can add a completely independent, purely oscillating set of modes for (Z4, Θ4): Δkp Σ p Z4 = Θ4 = − 2 R (b4 e−ipζ + ¯b4 eipζ ) , Δkp h b4 e−ipζ (Ω(2) + ir sin θ Ω(1)) + ¯b4 eipζ (Ω(2) − ir sin θ Ω(1))i . Regularity of the metric and dilaton factors mean that one should have ZA > 0 for A = 1, 2. This means that the terms in the parentheses in (4.5) must be strictly positive and since |Δ| < 1 away from the source, one can certainly guarantee ZA > 0 by taking: HJEP05(216)4 |bA| ≤ 2 , 1 A = 1, 2 . One may be able to improve this bound slightly, but the important point is that |bA| will always be bounded by a number of order 1. The second layer of equations Consider a single mode of ω and F : ω = e−iqζ (ωˆrdr + ωˆθdθ + ωˆ1dϕ1 + ωˆ2dϕ2) , F = − W e−iqζ (4.11) then the differential operators in (2.20) and (2.21) may be written as: Dω + ∗4Dω + F Θ3 ≡ e−iqζ (r2 + a2) cos θ Ω(1) (q) + r sin θ Ω(3) (q) + L1 L3 L2 cos θ Ω(2) (q) , (4.12) (q) L0 ≡ Σ r ∂r(r (r2 + a2) ωˆr) + 1 sin θ cos θ ∂θ(sin θ cos θ ωˆθ) + (r2 +a2) ωˆ1 + i kq a2 i kq cos2 θ i kq L1 ≡ (∂rωˆθ − ∂θωˆr) − r(r2 +a2) sin θ cos θ (r2ωˆ1 − a2 sin2 θ ωˆ2) , r L2 ≡ cos θ ∂rωˆ2 + (r2 +a2) sin θ ∂θωˆ1 − Σ cos θ ωˆr − L3 ≡ sin θ ∂rωˆ1 − r cos θ ∂θωˆ2 − Σ cos θ (a2 sin θ cos θ ωˆr − r ωˆθ) + 4kRa2r sin θ Σ2 W , k2q2(r2 +a2 sin2 θ) (r2 + a2) cos2 θ i kq 1 (q)W ≡ Σ r ∂r(r(r2 +a2)∂rW ) + sin θ cos θ ∂θ(sin θ cos θ ∂θW ) − Using the solutions in (4.5)–(4.9), the source terms in (2.20) and (2.21) give rise, a priori, to four non-trivial kinds of source terms: and have singularities involving Σ−2. and have singularities involving Σ−2. 1(a). Terms arising from products of modes with same phase. These depend upon e±2inζ 1(b). Terms arising from the product of a mode and a QΣA term. These depend upon e±inζ 2. Terms arising from the product of a mode and the constant (1) in ZA. These depend upon e±inζ and have singularities involving Σ−1. 3. Terms arising from product of modes with the opposite phase. These are independent of ζ and have singularities involving Σ−2. However, the sources of types 1(a) and 1(b) are not really distinct in that the solution is the same but simply with a different mode number. We therefore break down the sources into types 1,2 and 3 and write the particular equations that need to be solved and find the (4.19) (4.20) (4.21) (4.22) h , (4.23) particular solutions. Source 1: These systems of equations are: i q q L(1q) = − 2 R Σ(r2 + a2) cos θ Δkq r sin θ L(2q) = − 2 R Σ(r2 + a2) cos θ Δkq r , , Source 2: Source 3: i q q L(1q) = − 2 R (r2 + a2) cos θ Δkq r sin θ L(2q) = − 2 R (r2 + a2) cos θ Δkq r , , L(3q) = 0 , Lb(q) W − iRq L(0q) = q 2 Δkq 2 R2 Σ2 , L(3q) = 0 , Lb(q) W − iRq L(0q) = q 2 Δkq 2 R2 Σ , L1 (q=0) = 0 , L2 (q=0) = − R Σ(r2 + a2) cos θ , m Δ2mk r L3 (q=0) = 0 , Lb(q=0) W = Δ2km m2 R2 Σ(r2 + a2) cos2 θ , These equations have a gauge invariance associated with changing the u-coordinate: u → u + f (xi, v) , ω → ω − df + β ∂vf , F → F − 2 ∂vf . In terms of the qth mode this becomes (ωˆr, ωˆθ, ωˆ1, ωˆ2; W ) → (ωˆr, ωˆθ, ωˆ1, ωˆ2; W ) + ∂rh, ∂θh, i kq a2 sin2 θ Σ h, for an arbitrary function h(xi) on the base, B. In particular, for q 6= 0 one can choose a gauge with W = 0. It is relatively easy to find the explicit solutions for each of these sources: – 19 – tem [11, 30, 47, 48, 78]. the decoupling limit are Instead of trying to carry through the somewhat daunting task of determining the non-linear corrections to the supergravity-CFT map, we will instead proceed somewhat differently, and determine what strands are present in the CFT (and in what amounts) by an analysis of the two-charge solutions on which the coiffured solutions are based. Information from two-charge solutions. Our proposed dual CFT states involve frac tional spectral flow on a two-charge 14 -BPS state, and spectral flow does not change the strand content of a BPS state. Therefore, we can determine the amounts of the various types of strands present (at the fully non-linear level) in both styles of coiffuring, by studying the known map between CFT and supergravity for the two-charge sys The harmonic functions determining the geometry of a circular F1-NS5 supertube in Z2 = Z1 = A = − L β = −A√+ B , 2 L L Q2 Z L 1 0 |xi − gi(vˆ)|2 dvˆ , Q2 Z L |g˙i(vˆ)|2 + |g˙5(vˆ)|2 dvˆ , 0 |xi − gi(vˆ)|2 Q2 Z L g˙j(vˆ) dxj 0 |xi − gi(vˆ)|2 dvˆ , ω = −A√− B , 2 dual with respect to the flat transverse R4 parametrized by xi. where the dot on the profile functions indicates a derivative with respect to vˆ and ∗4 is the The onebrane charge is given by Q1 = Q2 Z L L 0 |g˙i(vˆ)|2 + |g˙5(vˆ)|2 dvˆ . Q1 = (2π)4 n1 gs2 α′3 V4 , Q2 = n5 α′ , The quantities Q1, Q2 are related to quantized onebrane and fivebrane numbers n1, n5 by where V4 is the coordinate volume of T4. The circular supertube profile is is given by g1 + ig2 = a exp[2πikvˆ/L] . It will prove convenient to denote x = x1 + ix2, y = x3 + ix4, and parametrize the profile by ξ ≡ 2πkvˆ/L. Since the supertubes of interest run around the same profile k times, the integral is simply k times the integral over the range ξ ∈ (0, 2π). The further change of variables z = eiξ, and the use of z¯ = 1/z for an integral along the unit circle in z, converts the integrals into contour integrals for which we can use the method of residues, for example Z2 = Q2 I dz 2πi 1 (5.45) (5.41a) (5.41b) (5.41c) (5.41d) (5.42) (5.43) (5.44) The poles in the integrand are located at where w˜ = xx¯ + yy¯ + a2, and so z ± = w˜ ± √w˜2 − 4xx¯a2 , Z2 = √w˜2 − 4xx¯a2 leads to the correct form of Z2 in the decoupling limit, x = r˜ sin θ˜eiϕ1 , y = r˜ cos θ˜eiϕ2 r˜ = pr2 + a2 sin2 θ , cos θ˜ = Σ Converting from Cartesian coordinates to spherical bipolar ones HJEP05(216)4 Next, we introduce ν = kp for convenience and we add a g5 term to the profile function, −b4 (zν − z−ν ) , where b4 is real, and corresponds to the magnitude of the quantity b4 in the supergravity (4.8). The quantity that corresponds to the phase of the supergravity b4 is a shift in vˆ in (5.50). In what follows, for both Style 1 and Style 2, we will take b4 to be real, both for convenience and for ease of comparison to [21]. This g5 term in the profile function gives rise to the following contour integral expression for the harmonic function Z4: The zν term yields the result The denominator gives again the factor of Σ; furthermore, one has Z4 = b4 2πi 1 I dz zν + z−ν b4 z ν − (−ax¯)(z− − z+) . One can then rewrite zν as − z −ν = z z+ − = The harmonic function depends on the combination sin θ eiϕ1 rather than cos θ eiϕ2 because the linearized supergravity modes getting a vev correspond to chiral rather than (5.46) (5.47) (5.48) (5.49) (5.50) (5.51) (5.52) (5.53) (5.54) twisted-chiral operators in the CFT. The fractional spectral flow operation converts one to the other. One is looking to match the structure in [21] equation (3.11c) which is this is exactly what is found once the contribution from the z−ν term in (5.51) is added. So the seed Z4 is Z4 = 2b4 Style 2. For coiffuring Style 2, the harmonic function Z1 exhibited in [21] equation (3.11a) for the corresponding two-charge seed solution, translated into our conventions is Z1 = Q1 + Σ 2b24 Q2 which follows from equation (5.41b). Thus we see that Style 2 coiffuring is the result solely of exciting |00i strands of length κ = kν = k2p; the strength b1 ∝ b42 of the vev is entirely accounted for by non-linear effects of the |00i strands, and so no additional contribution corresponding to nonzero n3,nˆ, n4,mˆ is necessary. The corresponding two-charge solution is precisely as in [21], and the spectral flow that adds the third charge simply turns vevs from chiral to twisted-chiral — under fractional spectral flow, the factor sinν θ eiνϕ1 turns into cosν θ eiνϕ2 , which is what we see in the coiffured harmonic functions of section 4 above. strands, the deformation profile becomes Style 1. It remains to match Style 1 to a set of supertube strands in a two-charge solution prior to spectral flow and coiffuring. We expect to at least have strands of length k(kp + c) for c = {−1, 0, +1}, since vevs for the operators (5.35)–(5.37) must appear at linear order in b4. The c = ±1 strands correspond to |±±i cycles and so in general will affect the location of the supertube profile in the transverse R4. Introducing |++ik(kp+1) and |−−ik(kp−1) g1 + ig2 = az + b−z−(kp−1) + b+zkp+1 = z a + b−z−ν + b+zν , (5.58) where ν = kp and z = exp(2πikvˆ/L). For small amplitude deformation b ± ≪ a, there are no new poles inside the contour of integration, and the pole in the integrand will still be close to z . We can map the profile back to a unit-velocity circular profile (i.e. g1 + ig2 = aeiw) via a single-valued conformal − map, at the cost of a Jacobian for the transformation. In general this leads to an infinite series in the expressions for Z1 and Z2 if there are only one or two lengths of strand in the profile g1 + ig2; since we wish to engineer a finite Fourier series for Z1 and Z2, the dual CFT state will have a series of strand lengths involving all possible multiples of p. Working firstly to leading order in b±, consider the profile ξ(vˆ) = 2πkvˆ L , w(ξ) = ξ − ν b cos νξ + . . . (5.59) where we set b+ = b − ≡ −iab/2ν in order that the map is a proper element of Diff(S1). Here again ξ serves as a rescaled periodic coordinate which ranges over [0, 2πk). The motivation for considering such a profile comes from coiffuring — the idea is that coordinate transformations on the supertube worldvolume apply a density perturbation to the round supertube without perturbing its location in space. The fivebrane and onebrane charge densities will no longer be constant along the supertube. Expanding this profile out to leading order in b reproduces (5.58). Such a change of variable has no effect on Z4 (which is reparametrization invariant) but it will change Z1 and Z2. The integration measure picks dw dξ dξ dvˆ (5.60) where we have expanded the Jacobian factor to clarify that w means w ξ(vˆ) as above. Similarly the “energy densities” in the numerator of Z1 in (5.41) pick up a factor dw 2 dvˆ |(g˙1 + ig˙2)(vˆ)|2 + |g˙5(vˆ)|2 → |(g1′ + ig2′)(w)|2 + |g5′(w)|2 (5.61) where primes denote derivatives with respect to the argument. Again evaluating the factors of (dw/dvˆ) to leading order in b one finds that in the new integration variable w, the integrand of Z2 is modified by a factor of 1 − b sin νw in (5.41a), while the integrand of Z1 gets a factor of 1 + b sin νw (in addition to the corresponding factors of 2πk/L). We then find the same sort of integral we encountered in (5.41b), with the same result. Our primitive approximations give b1 = −b2, which is appropriate for a2 ≪ Q1, Q2; the latter is a consequence of the decoupling limit. In principle one can proceed order-by-order in a series expansion in bI , (I = 1, 2, 4), working out the non-linear map between the vˆ coordinate frame in which the strand content is specified, and the w coordinate frame in which the supertube profile is a constant velocity parametrization of a circle. However, ultimately we are interested in the harmonic functions ZI having a single non-trivial Fourier coefficient. In Style 1, the perturbation to Z2 looks like (5.51) with ν = kp and a coefficient b2; and Z4 is the same but with a coefficient b4. These simple forms suggest that the more straightforward route is to work directly in the w coordinate frame and only implicitly specify the coordinate map via its inverse, Plugging this into (5.41a) gives exactly the right result for Z2 and Z4 in Style 1, using sin ν w ξ(vˆ) . (5.63) b ν 2b4 We have now used up almost all our freedom to specify the state; all that remains are the amplitudes b, b4. The integral for Z1 is Z1 = = k2Ry2 Z 2π 2πQ2 0 k2Ry2 Z 2π 2πQ2 0 dw dw dw −1 |(g1′ + ig2′)(w)|2 + |g5′(w)|2 (dw/dξ)2 dξ |x − (g1′ + ig2′)(w)|2 + |y|2 a2 + (2b4/kRy)2 cos2 νw 1 − b sin νw 1 |x − aeiw|2 + |y|2 (5.64) where we have used the relation Let us choose L = where the second equality is the analog of the gravity regularity condition (4.55). Then the factor in the integrand becomes = (a2 + (2b4/kRy)2) 1 + b sin νw = Qk21RQy22 1 + b sin νw , (5.67) where the last equality comes from evaluating the expression (5.42). All harmonic functions have only terms that are constant or a single S3 harmonic of the form (5.54), with m = n = p a 2 and b1 = −b2 = ib4/ ≪ Q1, Q2 of the amplitude relations (4.41) of Style 1 coiffuring in supergravity. For these two-charge solutions, the mode amplitude restrictions do not come from requiring regularity of the supergravity solution — all the two-charge solutions are nonsingular. Rather, the restriction comes from the somewhat arbitrary requirement that the harmonic functions contain only a single Fourier mode rather than a combination of modes √Q1Q2. These results agree precisely with the decoupling limit of different wavenumbers. It is worth reiterating that the map between supergravity and CFT takes place in the vˆ coordinate frame, which is only implicitly specified above through the relation (5.62). In the vˆ coordinates the solution is very complicated and has in principle all values of mˆ, nˆ, pˆ turned on. The non-zero values of mˆ, nˆ are given by the non-zero Fourier coefficients of (g1 + ig2)(vˆ) = a exp[iw(ξ(vˆ))], (5.65) (5.66) (5.68) (5.69) cn = Z 2π dξ 0 2π e−inξ eiw(ξ) = e−inξ(w)+iw Z 2π dw dξ 0 2π dw and are predominantly concentrated on the lowest modes. Expanding in b, cn = Z 2π dw 0 2π 1 + b cos(νw) ei(1−n)w X∞ 1 ℓ=0 ℓ! −inb cos νw ℓ ν one sees that the only nonzero Fourier coefficients occur for n = 1 + qν, q ∈ Z, generalizing (5.58). Of these non-zero Fourier coefficients, the positive values of n give the non-zero values of mˆ , and the negative values of n give the non-zero values of nˆ. Thus we see that mˆ and nˆ must be multiples of p, the mode number of the supergravity solution. Similarly, the non-zero values of pˆ are all multiples of p. In addition, reality of the conformal map implies c1+qν = c∗1−qν , which in turn means that for each q, the average numbers of |++ik(qkp+1) and |−−ik(qkp−1) strands are equal. Finally, note that in the quantum theory, there is a maximum mode number N = N1N5 and so one cannot precisely generate Style 1 because the Fourier expansion is necessarily finite; the result will differ at the 1/N level. For kp = 1, this family of states has a somewhat degenerate limit, since the length of the |−−ik(kp−1) strands is zero. This simply means that this particular strand type is absent for kp = 1, while the other strands remain as described above. In particular, the average numbers of |++ik(qkp+1) and |−−ik(qkp−1) strands are equal for q ≥ 2. Summary of proposed dual CFT states. In both Style 1 and Style 2, we start with a two-charge seed solution, determined by a profile function. The general dictionary for two-charge states is discussed in [30, 47, 48, 77]. We now describe how it applies to our twocharge seed states. Given a profile function, the non-zero Fourier coefficients specify the types of strands involved in the dual CFT state, and the values of the Fourier coefficients control the coefficients of the individual terms in the coherent state superposition. Style 2 profile function (5.44), (5.50): As shown in the previous subsection, the Style 2 seed solution is determined by the (g1 + ig2)(vˆ) = a exp i vˆ , 2πk L Since both g1 + ig2 and g5 have only a single Fourier mode, the dual CFT state contains just two types of strands, |++ik , |00ik2p , where the excited strands are only of one type, given by pˆ = p. To form the coherent state, one considers all partitions of the N1N5 copies of the CFT into strands of the two above types. Then one forms a sum in which the coefficients of these different partitions are controlled in a specific way by the two non-zero Fourier coefficients of the profile function (5.70) (for more details, see in particular the discussion in [77]). Given this seed two-charge state, we excite all strands except for the |++ik strands in the way described in section 5.4, so that the resulting three-charge state is composed only of strands of type |++ik , (J−+1/k)kp|00ik2p . The coefficients in the coherent state sum remain as in the two-charge seed solution. Style 1 For Style 1, the seed solution is given by the profile function (5.71) (5.72) (g1 + ig2)(vˆ) = a exp iw ξ(vˆ) , sin ν w ξ(vˆ) (5.73) where from (5.62) we specify the map implicitly through its inverse, ν Since both g1 + ig2 and g5 have an infinite Fourier series, the types of CFT strands present are those of type where mˆ, nˆ, pˆ can be any independent multiples of p, compatible with the total number of strands being N1N5. The coherent state has many more ingredients, however the coefficients in the superposition are again fully specified by the Fourier coefficients of the profile function (5.73). Therefore all the coefficients are determined by the parameter b4 (since b4 Given this seed two-charge state, we again excite all strands except for the |++ik strands in the way described in section 5.4, so that the resulting three-charge state is composed only of strands of type |++ik , (J−+1/k)kpˆ|00ik2pˆ , (J−+1/k)knˆ|++ik(knˆ+1) , (J−+1/k)kmˆ |−−ik(k mˆ−1) , (5.76) where again the values of mˆ, nˆ, pˆ are independent multiples of p, compatible with the total number of strands being N1N5, and the coefficients in the coherent state sum remain as in the two-charge seed solution. Finally, note that, at the level of counting free parameters in the solutions, we expect there to be good agreement more generally between coiffured deformations of circular supertubes on the supergravity side, and fractional spectral flows of circular two-charge seed solutions on the CFT side. On the CFT side, one has two functional degrees of freedom — the specification of the profile of the |00i strands embodied in the function g5, and the diffeomorphism w(ξ) that changes the parametrization of the round supertube. On the supergravity side, the diffeomorphism w(ξ) corresponds to the charge densities ̺1 and ̺4 discussed in section 3.2, and the profile of the |00i strands corresponds to the function Z4. In section 3.2 we saw that in the absence of Z4 there are three functions and two functional constraints, leaving one functional degree of freedom; adding in Z4 gives two functional degrees of freedom, which agrees with the CFT. There are interesting parallels between the supergravity construction of section 3 and the appearance of density fluctuations in the CFT. However the relationship is not direct. In the CFT, the density profile appears in the two-charge seed solutions before applying fractional spectral flow; on the gravity side, the density perturbations were introduced in a spectrally inverted frame, and then a second spectral inversion was applied to transform back to the original frame. The density fluctuations were thus applied to a supertube that does not have a simple, direct relationship to the original D1-D5 CFT. There is also the technical distinction in that the construction of section 3 initially involves three apparently independent charge density functions that must then satisfy the constraints of supertube regularity (3.19) and (3.20), leaving only one independent density function. In this section, the density fluctuation is introduced via a combination of the g5 profile and (in Style 1) a conformal map of the round supertube profile, which, via the Lunin-Mathur map, automatically maintains the supertube regularity conditions. It would be very interesting to investigate this relationship in more detail. 5.5 We now compare the angular momenta J 3, J¯3 and the momentum charge QP , and demonstrate the agreement between our supergravity solutions and our proposed dual CFT states. For ease of comparison to the supergravity discussion in section 4.4, we revert to the D1-D5 duality frame. − k The discussion that follows requires a certain amount of notation to write the charges explicitly, however the reasons that underlie the agreement can be stated simply. Firstly, all our momentum excitations can be expressed in terms of the action of powers of J +1 . Secondly, for each pˆ the average numbers of the strands of length k2pˆ + k and k2pˆ − k are equal, because of their origin as the (real-valued) two-charge density profile. Therefore adding momentum pˆ requires, on average, k2pˆ strands of the CFT. This fact leads to the relation between the angular momenta J 3, J¯3 and the momentum charge QP observed in the supergravity, as we now show explicitly. Style 2. For Style 2, we have a coherent state which is a sum of terms of the form where the sum runs over all n2 such that |++ik , kn1 + (k2p)n2 = N1N5 , weighted with coefficients as described in the previous subsection. For Style 2 coiffuring, from (4.82) and (4.86) we have on the gravity side (5.77) (5.78) (5.80) (5.81) (5.82) (5.83) (5.84) QP = 2|b4|2 , k2Ry2 JR = 2 kRy 1 kRyQP , JL = JR + kRyQP . (5.79) In the CFT, the expectation value of the momentum L0 − L¯0 in the Style 2 state is The total number of strands is N1N5; this determines n¯1 in terms of n¯2 (or Np) as Np = p n¯2 . n¯1 = N1N5 k k k Np . Then the CFT ¯3 is We convert the supergravity charges to quantized charges using Q1 = , which lead to the useful relations Q5 = gsN5α′ , QP = gs2NP α′4 Ry2V , π 4G(5) = V Ry gs2α′4 4G(5) Ry Q1Q5 = N1N5 , π 4G(5) RyQP = NP . Thus we obtain which agrees with the CFT. Next, the CFT j3 is Comparing to the gravity solution we have 4G(5) JR = 2 n¯1 + (kp)n¯2 = ¯3 + kNp . 3 jgrav = 4G(5) JL = ¯g3rav + kNp n1 Y pˆ∈pZ QP = which is also in agreement. Then by comparing the momentum charge we obtain the map between |b4|2 and n¯2: NP = 4G(5) RyQP p n¯2 = 4G(5) 2|b4|2 . k2Ry Style 1. For Style 1, we first consider kp > 1. As described above, the ingredients in the coherent state sum are |++ik For Style 1 coiffuring, from (4.57), (4.66) and (4.67) we have on the gravity side 4|b4|2 , k2Ry2 JR = 2 kRy 1 kRyQP , JL = JR + kRyQP . In the CFT, each individual element in the coherent state sum has momentum eigenvalue X pˆ(n2,pˆ + n3,pˆ + n4,pˆ) Np = pˆ(n¯2,pˆ + n¯3,pˆ + n¯4,pˆ) . n¯3,pˆ = n¯4,pˆ for all pˆ. (5.85) (5.86) (5.87) (5.88) n4,pˆ (5.89) (5.90) (5.91) (5.92) (5.93) (5.94) and so the expectation value of L0 − L¯0 again involves the average numbers of strands, Since the total number of strands is N1N5, we have kn¯1 + X pˆ∈pZ (k2pˆ)n¯2,pˆ + (k2pˆ + k)n¯3,pˆ + (k2pˆ − k)n¯4,pˆ = N1N5 . Because the density profile function w(ξ) is real, we have the relation on the average numbers Therefore we have and so, as for the Style 2 states, n¯1 is given by kn¯1 + X (k2pˆ) (n¯2,pˆ + n¯3,pˆ + n¯4,pˆ) = N1N5 n¯1 = N1N5 − kNp . The CFT j3 is ¯3 = 1 2 n¯1 + X (n¯3,pˆ − n¯4,pˆ) = n¯1 = 2 k − 2 Np . k − 2 Np , j3 = 2 n¯1 + X (n¯3,pˆ − n¯4,pˆ) + X (kpˆ) (n¯2,pˆ + n¯3,pˆ + n¯4,pˆ) = ¯3 + kNp . (5.99) (5.95) (5.96) (5.97) (5.98) (5.100) HJEP05(216)4 Comparing to the gravity we have 3 jgrav = 4G(5) JL = ¯g3rav + kNp and so we again find perfect agreement. average numbers of excited CFT strands, Finally, by comparing the momentum charge we obtain the map between |b4|2 and the NP = 4G(5) RyQP X pˆ(n¯2,pˆ + n¯3,pˆ + n¯4,pˆ) = pˆ∈pZ π 4|b4|2 . 4G(5) k2Ry (5.101) For kp = 1, the analysis contains minor differences, however the expressions for j3 and ¯3 in terms of N1, N5, NP are the same as those given in (5.98) and (5.100), as we show in appendix C. Thus the conserved charges agree for all values of k and p. Therefore we find exact agreement of conserved charges between gravity and CFT, providing supporting evidence for our proposal. It would be interesting to scrutinize our proposal further with the tools of precision holography [30, 47–49, 77]. 5.6 Comments on momentum fractionation The fractional spectral flow that we perform results in filled Fermi seas on the excited strands. One way to see this is to observe that the SU(2)R current algebra has the identity (J−+1/k)kpˆ |00ik2pˆ = J +2kpˆ−1 · · · J−+k25pˆ J + − k2pˆ − k23pˆ J + − k21pˆ |00ik2pˆ . (5.102) Similar expressions apply for the |++i and |−−i strands. So the CFT state can be written in different ways, and in one way of looking at our states, we excite modes with the lowest possible energy compatible with the constraint of integer momentum per strand. Saying this another way, spectral flow creates a state with the lowest possible energy for a given angular momentum, or equivalently maximal angular momentum for a given energy, so that there is no available free energy for thermal excitations of the state. As one backs away from maximal angular momentum, one has the freedom to excite different modes, and the entropy increases. For instance, if we change one of the current raising operators on the right-hand side of (5.102) from a J + to a J 3, the angular momentum is decreased by one unit but the energy and momentum remain the same; and there are kp distinct ways to do this. Decrease the angular momentum by one more unit, and we can either have one J − or two J3 with the rest remaining J +, and there are of order (kp)2 choices; and so on. Such a deformation away from maximal spin preserves the BPS property of the CFT state. It is interesting to ask what the gravitational description of such excitations will be, and whether they will match those of the CFT. If we change the lowest modes with energy/momentum of order 1/k2p, we would expect to have made a change in the geometry in the places with the deepest red-shift. Note that such BPS deformations are not available in the two-charge seed on which the three-charge coiffured solution is based. Since the CFT state has strands of length of order k2p, there are also non-BPS excitations that have zero momentum and angular momentum, and energy of order 1/k2p. Such excitations are also present in the two-charge seed states. In the supergravity, the non-BPS excitations are described at the linearized level by solving wave equations in the superstratum geometry. The supergravity solutions do not appear to have excitations at the scale 1/k2p suggested by the CFT, however; in general, there seems to be a mismatch between the gap in supergravity and in the CFT. The two-charge seed for Style 2 coiffuring is quite similar to a class of two-charge solutions studied in [29], for which the gap was estimated to be a/b (with a related to the number of |++i strands, b the number of T4 strands including |00i strands). In the CFT, the gap depends only on the length κ of the strands and is independent of the relative amounts a and b of the different kinds of strands. A preliminary study of the foregoing three-charge geometries indicates that, similar to the examples of [29], the red-shift depends on the amplitudes a and b, and that the deepest red-shifts are not kp times deeper than those of the parent k-wound supertube. In general, one can arrange that the throat in supergravity is deeper and results in a smaller gap than in the orbifold CFT (e.g. supergravity duals to CFT states discussed in [29] having only short cycles but low total angular momentum), and in yet other examples the throat in supergravity is shallower and results in a larger gap than in the orbifold CFT (e.g. the coiffured geometries discussed in this paper when b is finite but much less than a). It would be useful to understand better the cause of this discrepancy. The two-charge seed geometries of section 5.4 offer a qualitative explanation of the gap in supergravity. The dual of the F1-P source in the Lunin-Mathur construction of two-charge geometries [79] is a D1-D5 supertube smeared over the compact directions — the circle parametrized by y and the compactification manifold M [11, 29, 30, 78]. When segments of the unsmeared source approach one another, a throat opens and deepens in the geometry. This property explains why the profile (5.44) results in a red-shift of order k — HJEP05(216)4 the supertube source traces the same profile in the transverse space k times in the course of the supertube winding the y circle, and is k times more compact (in R4 coordinates); as a consequence, the harmonic functions are k times bigger at their maximum, and the throat is k times deeper. For a small perturbation of this profile, it may be that the oscillations of the profile are kp times faster than the k-fold spiral of the supertube, but this is a small perturbative wiggle and does not make the profile kp times more bunched together, and hence the deepest parts of the throat do not exhibit a red-shift kp times deeper. However, as one shifts more of the strands from |++i type to |00i type, the angular momentum is reduced, the source becomes more compact, and the throat deepens. It remains a puzzle why there is such a mismatch between the behavior of supergravity and that of the CFT for such a coarse property of the geometry. The gap to non-BPS excitations is of course not a robust property of the system, and could change dramatically as one passes from the regime where the CFT is weakly coupled to the regime where it is strongly coupled and gravity is a good approximation. Nevertheless, there are examples (see for instance [53]) where the gap can be matched on both sides of the duality. The presence or absence of strands polarized in the T4 directions appears to be an ingredient which influences whether this quantity agrees between gravity and CFT; it would be useful to understand fully when this comparison does and does not work. 6 Discussion This work has expanded the construction of superstrata to include momentum-carrying modes in deep AdS3 throats, in which the red-shift at the bottom of the throat is k times that of a singly-wound supertube. Our construction started from a k-wound circular supertube geometry. We performed spectral inversion on this solution, then altered its angular momentum by adding charge density fluctuations along the supertube with a wavenumber kp for some integer p, without deforming the shape of the supertube. We then brought the solution back to the original frame, where these fluctuations became momentum-carrying Our construction also produced the first examples of asymptotically-flat superstrata. We built two classes of solutions, corresponding to two different ways of arranging the Fourier coefficients in order to obtain smooth solutions (with the usual Zk orbifold singularities at the location of the supertube). Taking the decoupling limit to obtain the corresponding asymptotically-AdS solutions, we derived a proposal for the dual CFT states, for both classes of solutions. The starting supertube is built from a macroscopic ensemble of cycles of length k in the twisted sector of the symmetric orbifold CFT. The angular excitations in the CFT description are coherent fractional spectral flows on additional cycles of the twisted sector state, whose length is of order k2p. This fractional spectral flow can also be thought of either as acting of order kp times with the fractionally-moded raising operator J−+1/k, or as raising the Fermi seas on these cycles by filling all the fermion modes with positive R-charge up to a level of order 1/k. In our states, the fractionally-moded quanta in the CFT correspond to perfectly regular, local excitations in the supergravity theory and not to non-geometric or multi-valued perturbations. The bulk reflection of the fractional momentum carriers is rather the redshift of the perturbations down the supertube throat. A small puzzle that remains is the apparent mismatch in the excitation gap of orbifold CFT states and supergravity geometries discussed in section 5.6. A very similar mismatch was previously noted for certain two-charge solutions [29]. In the CFT, the gap is determined by the length of the longest cycles in the twisted sector ground state. In the geometry, the depth of the throat depends on other quantities, such as the relative proportions of the different strands. The supergravity gap can be larger or smaller than the orbifold CFT gap. The gap to non-BPS excitations is not protected in general, so this is not a serious problem for the holographic duality. However there are examples (see for instance [53]) where the gap matches between gravity and CFT. It would be interesting to understand when the gap should agree, and when it should not. Our solutions do not have all desired features of typical black-hole microstates: their angular momenta are over-spinning and the throats are not as deep as those of typical states. The corresponding orbifold CFT states contain strands having length of order k2p, and so k can at most be of order √N1N5, while the longer wavelength scale k2p is not apparent in the geometry. Thus we regard the supergravity solutions presented here as a “proof of concept” of a supergravity realization of momentum fractionation on superstrata, much like the solutions in [21] are a proof of concept of the existence of superstrata solutions parameterized by arbitrary functions of two variables. For the future, one would like to improve on both of these (related) features: to lower the angular momenta, and to deepen the throat further. First, regarding the angular momenta, in section 5.6 we identified CFT excitations that move away from the maximally spinning/overspinning regime by reducing the angular momentum through a change in the polarization of the R-symmetry currents acting on the two-charge seed. Using this freedom, one can make available some of the free energy to wiggle the throat while remaining BPS. Where in the throat the excitation lies should correlate with the degree of fractionation of the modes whose polarizations are being adjusted in the CFT. One place to look for these more general solutions on the supergravity side is to consider more generic superstrata, described by arbitrary functions of two variables. In this work we have focused on a sub-class of solutions which are parameterized by functions of HJEP05(216)4 one variable. This has been a choice made for technical convenience, to focus on the physics of momentum fractionation in a tractable system. It would be interesting to generalize our solutions to superstrata which are parametrized by functions of two variables and which exhibit momentum fractionation. Looking further ahead, the generic CFT state deformations discussed above, which stay BPS by deforming the polarizations of the spectral flow R-currents, will correspond to deformations of the supergravity solution that depend on all all three angular variables (v, ϕ1, ϕ2). The next essential step in the study of superstrata is to construct states with deeper throats, that are in a macroscopic scaling regime. Our solutions have throats k times deeper than the first superstrata constructed in [21], and so represent progress in this direction. The standard way to obtain a macroscopic scaling solution is to use at least three Gibbons-Hawking centers, but it may also be possible to construct scaling solutions with two centers when the supertubes fluctuate. As we noted above, for technical reasons we have focussed on some very particular modes and this choice of modes meant that whenever we added momentum to the supertube we also added a similar amount of angular momentum. Thus our solutions remained over-spinning or extremal. As a result, we could not access the scaling region that is usually associated with the microstates of a black hole with macroscopic horizon area. In this paper we added charges to the supertube in a manner that precluded us from exploring such deep, scaling geometries. In addition to the excitations discussed above that lower the angular momenta, more broadly one can consider excitations that either have no angular momentum, or have negative angular momentum. In principle, by using these excitations one can add momentum to the supertube in a way that takes the charges into the BMPV regime. The corresponding black hole would then have a macroscopic horizon and the microstate geometry should then scale and exhibit larger red-shifts and lower holographic energy gaps. This is presently under investigation. More generally, one may desire to embed superstrata and the kind of twisted-sector structure elucidated here, in multi-centered deep, scaling geometries since this is (as yet) the only known way to access typical twisted-sector CFT states within the supergravity approximation. On a technical level this will be challenging, since it means going beyond two centers and yet our construction has made very heavy use of the flat R4 base and the separability of various wave equations in bipolar coordinates. However, this does not mean that it is impossible: the scalar Green functions for charge density fluctuations in generic ambipolar backgrounds were discussed in [37], and a three-centered Green function was constructed explicitly. So while this may be very difficult, it is not completely out of reach. Moreover, we hope to find physical arguments that illuminate what the geometries constructed in this paper will probe once they are combined with generic superstrata and embedded in deep, scaling geometries. Looking further to the future, it would be of great interest to study momentum fractionation in non-supersymmetric microstates, as done in [54]. The recent construction of multi-bubble non-BPS black-hole microstate geometries [80] offers the prospect of progress in this direction. Acknowledgments We would like to thank Stefano Giusto, Rodolfo Russo and Masaki Shigemori for helpful discussions. The work of IB and DT was supported by 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 FQXiRFP3-1321 (this grant was administered by Theiss Research). The work of EJM was supported in part by DOE grant DE-SC0009924. The work of DT was supported in part by a CEA Enhanced Eurotalents Fellowship. The work of NPW was supported in part by DOE grant DE-SC0011687. For hospitality during the course of this work, EJM and NPW are very grateful to the IPhT, CEA-Saclay; IB, EJM, and DT thank the Centro de Ciencias de Benasque Pedro Pascual; and DT and NPW thank the Yukawa Institute for Theoretical Physics, Kyoto University. A The BMPV black hole To help establish normalizations, it is useful to give the standard BMPV black-hole metric [71] in terms of the Ansatz used in this paper. Everything is, of course, v-independent and the vector field, β, and the ΘI , are set to zero. For a BMPV black hole located at the center of space (r = 0, θ = 0) the ZI are appropriately-sourced harmonic functions: Q1 Λ , J Z4 = 0 . (A.1) Q2 Λ , F = − Λ , 2Q3 The angular momentum vector, ω, is then simply the “harmonic” solution to the homogeneous equation (2.20) with source at the center of space: ω = Λ2 (r2 + a2) sin2 θ dϕ1 − r2 cos2 θ dϕ2 . (A.2) (A.3) (A.4) Note that as r → ∞ one has ZI ∼ 1 + QI r2 , I = 1, 2, 3 , which determine the charges and angular momenta of the black hole. To make the asymptotic analysis of the metric in the vicinity of the center of space using more standard spherical coordinates in the infinitesimal neighborhood of r = 0, θ = 0, ω ∼ r2 sin2 θ dϕ1 − cos2 θ dϕ2 , one can simply take: becomes: r = λ sin χ , θ = cos χ . and expand to lowest order in λ. One then finds that the leading part of the metric J λ a (cos2 χ dϕ1 + sin2 χ dϕ2) 2 (cos2 χ dϕ1 + sin2 χ dϕ2)2 . (A.5) ds52 = pQ1Q2 λ2 + dχ2 + sin2 χ cos2 χ (dϕ1 − dϕ2)2 dλ2 + 2Q3 Q1Q2 + 1 − 2Q1Q2Q3 dv − 2Q3 J J 2 In particular, we see that with our normalizations one must impose the condition: J 2 ≤ 2Q1Q2Q3 2 JL ≤ Q1Q2QP , (A.6) where JL = J/√2 and QP = Q3. B Reduction to five dimensions There are two standard ways of reducing the six-dimensional solution, and the system of BPS equations [38–41], to the standard, five-dimensional analogs found in may references (see, for example, [13, 81]). These two choices of reduction come from different embeddings of the five-dimensional fields in the six-dimensional formulation; we summarize these two standard choices here. The five-dimensional BPS equations are: Z3 P 1/3 (B.1) (B.2) (B.3) (B.4) (B.5) (B.6) (B.8) Θ(I) = ⋆4 Θ(I) , ∇2ZI = 21 CIJK ⋆4 (Θ(J) ∧ Θ(K)) , dk + ⋆4dk = ZI Θ(I) . Our goal will be to take v-independent, six-dimensional solutions and compactify on an S1 fiber so that the system equations (2.14)–(2.16), (2.20) and (2.21) reduce to the five(dt + k)2 + √ Z3 hdy + (1 − Z3−1)(dt + k) + (β − k) i2 P ds42(B) . (B.7) Compactifying on the y-circle yields an overall warp factor of on the fivedimensional metric and leads to ds52 = − Z3 P − 3 (dt + k)2 + Z3 P 3 ds42(B) , 1 These identifications reduce the six-dimensional BPS system used in this paper directly to the canonical five-dimensional system; this is the origin of how we have chosen to normalize the flux fields like ΘI . However we have chosen the t, y coordinates (2.2), meaning that F → 0 at infinity, leading to a canonical embedding more closely associated with supertubes. We will now describe this in more detail. dimensional system. B.1 Reduction 1 Upon making the identifications 1 P F F = −Z3 , the six-dimensional metric is given by P which can also be written as P P F P P 2 This is the canonical choice if F never vanishes and in particular, when F → −1 at infinity. One can then write the metric (2.4) globally as ds62 = √ (du + ω)2 − √ dv + β + F −1(du + ω) 2 + P ds42(B) . u = t , v = t + y , k = ω , Θ3 = dβ , Z3 hdv + β − Z3−1(dt + k) i2 P ds42(B) , 1 Z3√P + √ P ds42(B) . (B.11) With these identifications one must make the following replacements and re-definitions and complete the squares in the metric as in (2.6) to obtain for the quantities defined in the body of this paper ΘI → 2 ΘI , I = 1, 2, 4 ; Θ3 = √ 2 dβ . ment ω = √2k − β. Doing this, the BPS equations (2.14)–(2.16), (2.20) and (2.21) reduce to the fivedimensional system (B.1)–(B.3). In particular, the terms arising from the constant in F = −2(Z3 − 1) cancel in (2.20) against the terms Dβ + ∗4Dβ arising from the replace Reduction 2 In this reduction we use the coordinates (2.2): Then as described in (2.5), we introduce k = ω + β , Z3 P 1 2 (B.9) (B.10) (B.12) (C.1) (C.2) (C.3) C The lowest Style 1 modes In this appendix we demonstrate the agreement of conserved charges for the lowest possible modes in Style 1, those with kp = 1, following the analysis for kp > 1 done in section 5.5. For kp = 1, the dual CFT state is a particular superposition of states of the Style 1 type (5.89), strands are equal for pˆ ≥ 2, As explained at the end of section 5.4, the average numbers of |++ik(kpˆ+1) and |−−ik(kpˆ−1) n¯3,pˆ = n¯4,pˆ for all pˆ ≥ 2 , while the excited |−−i strands that would be counted by n4,1 would have length zero, which does not exist. Therefore we set This means that the excited |++i2 strands that are counted by n3,1 are not balanced out by corresponding |−−i strands. Nevertheless, the conserved charges will work out properly, as we now show. n4,1 = 0 . n¯1 + X [pˆn¯2,pˆ + (pˆ + 1)n¯3,pˆ + (pˆ − 1)n¯4,pˆ] = N1N5 n¯1 + X [pˆ(n¯2,pˆ + n¯3,pˆ + n¯4,pˆ)] + n¯3,1 = N1N5 ¯3 = 1 2 n¯1 + X (n¯3,pˆ − n¯4,pˆ) = (n¯1 + n¯3,1) = 1 2 1 2 N1N5 − 2 Np , 1 in perfect agreement with the value of ¯3 computed from the gravity. n¯1 = N1N5 − Np − n¯3,1 . (C.5) (C.7) Next, the CFT ¯3 is The CFT j3 is j3 = 1 2 1 2 n¯1 + X (n¯3,pˆ − n¯4,pˆ) + X pˆ(n¯2,pˆ + n¯3,pˆ + n¯4,pˆ) (n¯1 + n¯3,1) + X pˆ(n¯2,pˆ + n¯3,pˆ + n¯4,pˆ) = ¯3 + kNp . which again agrees exactly with the value of j3 computed from the gravity. The momentum charge determines b4 just as for kn > 1, and so all conserved charges agree. This agreement shows that comparing conserved charges alone does not put any constraint on the value of n¯3,1. 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Iosif Bena, Emil Martinec, David Turton. Momentum fractionation on superstrata, Journal of High Energy Physics, 2016, 64, DOI: 10.1007/JHEP05(2016)064