M-theory superstrata and the MSW string

Journal of High Energy Physics, Jun 2017

The low-energy description of wrapped M5 branes in compactifications of M-theory on a Calabi-Yau threefold times a circle is given by a conformal field theory studied by Maldacena, Strominger and Witten and known as the MSW CFT. Taking the threefold to be \( {\mathbb{T}}_6 \) or \( \mathrm{K}3\times {\mathbb{T}}^2 \), we construct a map between a sub-sector of this CFT and a sub-sector of the D1-D5 CFT. We demonstrate this map by considering a set of D1-D5 CFT states that have smooth horizonless bulk duals, and explicitly constructing the supergravity solutions dual to the corresponding states of the MSW CFT. We thus obtain the largest known class of solutions dual to MSW CFT microstates, and demonstrate that five-dimensional ungauged supergravity admits much larger families of smooth horizonless solutions than previously known.

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M-theory superstrata and the MSW string

Accepted: June M-theory superstrata and the MSW string Iosif Bena 0 1 2 5 Emil Martinec 0 1 2 3 David Turton 0 1 2 5 Nicholas P. Warner 0 1 2 4 0 University of Southern California , Los Angeles, CA 90089 , U.S.A 1 5640 S. Ellis Ave. , Chicago, IL 60637-1433 , U.S.A 2 CEA, CNRS, Orme des Merisiers , F-91191 Gif sur Yvette , France 3 Enrico Fermi Inst. and Dept. of Physics, University of Chicago 4 Department of Physics and Astronomy, and Department of Mathematics 5 Institut de Physique Theorique, Universite Paris Saclay The low-energy description of wrapped M5 branes in compacti cations of Mtheory on a Calabi-Yau threefold times a circle is given by a conformal eld theory studied by Maldacena, Strominger and Witten and known as the MSW CFT. Taking the threefold to be T6 or K3 T2, we construct a map between a sub-sector of this CFT and a sub-sector of the D1-D5 CFT. We demonstrate this map by considering a set of D1-D5 CFT states that have smooth horizonless bulk duals, and explicitly constructing the supergravity solutions dual to the corresponding states of the MSW CFT. We thus obtain the largest known class of solutions dual to MSW CFT microstates, and demonstrate that ve-dimensional ungauged supergravity admits much larger families of smooth horizonless solutions than previously known. AdS-CFT Correspondence; Black Holes in String Theory - 1 Introduction 2.1 2.2 2.3 2.4 3.1 3.2 3.3 4.1 4.2 5.1 5.2 2 D1-D5-P BPS solutions and spectral transformations D1-D5-P BPS solutions Canonical transformations General spectral transformations Five-dimensional solutions and spectral transformations 2.5 Spectral transformations for v-independent solutions 3 The multi-wound supertube and mapping D1-D5 to MSW The multi-wound D1-D5 supertube con guration Spectral transformations of the supertube Mapping from D1-D5-KKM to M-theory 4 General SL(2; Q) transformations in six dimensions General spectral transformation of the metric functions General spectral transformation of the gauge elds 5 Constructing M-theory superstrata D1-D5-P Superstrata Transforming the superstrata 5.2.1 5.2.2 5.2.3 Transforming the metric quantities Transforming the uxes M-theory superstrata 6 An explicit example 6.1 The D1-D5-P superstrata 6.2 The M-theory superstrata 7 Comments on symmetric product orbifold CFTs 7.1 Review of the D1-D5 CFT 7.2 The D1-D5-KKM CFT 8 Discussion A Covariant form of BPS ansatz and equations B Circular D1-D5 supertube: parameterizations C Lattice of identi cations and fractional spectral ow { i { D.1 Coordinate conventions D.2 MSW maximal-charge Ramond ground state solution 41 42 1 Introduction For around twenty years there have been two well-known routes to describe the entropy of BPS black holes in some form of weak-coupling limit. The rst of these was done using the original perturbative, weak-coupling microstate counting of momentum excitations of the D1-D5 system [1]. The corresponding three-charge black hole, in ve dimensions, is obtained by compactifying D1 and D5 branes of the IIB theory on M either T 4 or K3, and adding momentum. When the size of the S1 wrapped by N1 D1 S1, where M is and N5 D5 branes is the largest scale in the problem, the low-energy physics is given by a superconformal eld theory (SCFT) with central charge 6N1N5. There is now strong evidence that there is a locus in moduli space where this SCFT is a symmetric product orbifold theory with target space M N =SN , where N = N1N5 [2]. This orbifold point is naturally thought of as the weak-coupling limit on the eld theory side, and it is far in moduli space from the region in which supergravity is weakly coupled (see for example the review [3]). The D1-D5 system has eight supersymmetries that, in terms of left- and right-moving modes, are N = (4; 4). Adding NP units of left-moving momentum breaks the supersymmetry to (0; 4). The state counting emerges from the ways of partitioning this momentum amongst the fundamental excitations of the SCFT, and the result matches the entropy of the three-charge black hole in ve dimensions. The D1-D5 black string in six dimensions has a near-horizon limit that is AdS3 The holographic duality between the strongly coupled D1-D5 CFT and supergravity on AdS3 S3 has been widely studied and is, as far as these things go, relatively well understood. In particular, one can often do \weak-coupling" calculations in the orbifold CFT S3. that can be mapped to \strong-coupling" physics described by the AdS3 S3 supergravity dual of this theory. The computation of BPS black hole entropy is one such example. However the holographic duality gives much more information than simple entropy counts, enabling the study of strong-coupling physics of individual microstates of the CFT, and their bulk descriptions (see, for example, [4{14]). The second approach is to use the Maldacena-Strominger-Witten (MSW) [15] \string." Here one starts with a compacti cation of M-theory on a Calabi-Yau threefold to ve dimensions. One then wraps an M5 brane around a suitably chosen divisor to obtain a (1 + 1)-dimensional string in ve dimensions. This breaks the supersymmetry to N = (0; 4) right from the outset. There is a (1+1)-dimensional SCFT on the worldvolume of this string and its central charge is proportional to the number of moduli of the wrapped brane.1 This 1The central charge can be computed in terms of the intersection properties of the divisor through a simple index theorem. { 1 { MSW string is wrapped around another compacti cation circle to obtain a four-dimensional compact bound state. One can add momentum excitations to the MSW string in a manner that preserves the (0; 4) supersymmetry, and the entropy of these excitations matches (to leading order) the entropy of the corresponding four-dimensional BPS black hole. However, despite multiple attempts over the past twenty years, the holographic description of the MSW black hole remains much more mysterious. The strongly-coupled physics of this CFT is described by weakly-coupled supergravity in an asymptotically AdS3 symmetric-orbifold CFT anywhere in the moduli space [16, 17]. The purpose of this paper is to construct a map between a large sub-sector of the MSW CFT and a large sub-sector of the D1-D5 CFT. Our construction has two steps. We rst construct a map between the Type IIB [global AdS3] S 3 T4 solution dual to the NS vacuum of the D1-D5 CFT (and Z orbifolds thereof) and the M-theory [global AdS3] solutions are expanded into three sets of Fourier modes, labelled by (k; m; n). Our map takes the modes with k = 2m to asymptotically AdS3 S2 solutions dual to momentumcarrying microstates of the MSW CFT. Since, in principle, D1-D5-P superstrata with generic (k; m; n) are described by functions of three variables,3 the restriction to enable the map to MSW CFT reduces this to functions of two variables. In practice, in this paper we have constructed solutions with k = 2, m = 1 and generic n and so our M-theory superstrata are parameterized by one integer n. Extrapolating to superpositions of two modes will give, by the linearity of the BPS equations of six-dimensional supergravity, smooth solutions parameterized by functions of one variable.4 Either way, we are able to build the largest known class of smooth microstate geometries for the MSW black hole. Precise dual CFT states for superstrata solutions have been identi ed [11{14], and our map indicates that there is a one-to-one map between the subset of these states that are eigenstates of the R-current JL3 corresponding to rotations around the Hopf ber of the S3, and a certain class of states of the MSW CFT. Given that the MSW CFT does not 2More precisely, beginning with a set of D1-D5 R-R ground states that are related to (Z orbifolds of) [global AdS3] S 3 T4 [18, 19], we construct a map to the MSW maximally-spinning Ramond ground state, which is related to [global AdS3] S 2 T6 by spectral ow. 3Smooth solutions with generic Fourier modes have not yet been explicitly constructed, and it is conceivable that interactions between such generic modes could introduce new, unanticipated singularities. This is why we are using the phrasing \in principle." 4Another way to build M-theory superstrata parameterized by a function of one variable is to impose the k = 2m condition on the superstrata constructed in [11]. { 2 { seem to have a weakly-coupled symmetric orbifold description, the fact that one can map a sector of this CFT into a sector of the D1-D5 CFT may provide leverage in analyzing some aspects of the MSW CFT, such as the set of protected three-point functions where two of the operators are heavy and one is light. The [global AdS3] S 2 T6 solution dual to the NS vacuum of the MSW CFT can be obtained as an uplift of a Type IIA con guration with two uxed D6 branes of opposite charges (the uxes give rise to D4 charges which uplift to the M5 charges of the M-theory solution). The geometric transition that employs uxed D6 (and anti-D6) branes to convert black holes and black rings into smooth, horizonless geometries was rst described in [20, 21]. A particular example of this was studied in [22] in which a uxed D6-D6 bound state the solution [22, 24, 25]. Uplifting this supergravity solution to M-theory gives rise to a singular supergravity PP-wave solution, carrying angular momentum along both AdS3 and S2; however, this naive extrapolation ignores the non-Abelian and non-linear dynamics of multiple D0-branes. A second way to add momentum charge is to add a gas of supergravity modes directly in M-theory, in the smooth [global AdS3] S 2 T6 solution. The entropy of this \supergraviton gas" [26] scales in the same way as the added D0 branes described above [25]. However the back-reaction of this supergraviton gas was not constructed. The solutions we nd are smooth M-theory geometries carrying the same charges as the foregoing ensembles of states, the D0's and the supergraviton gas, and so it is natural to think of our solutions as examples of fully back-reacted smooth geometries associated to the supergraviton gas, or to correctly-uplifted D0 branes. Indeed, the linearized limit of the superstratum modes is explicitly a supergraviton gas in [global AdS3] Hence, at least outside the black-hole regime of parameters, it may well be that some of S 2 our M-theory solutions are fully back-reacted supergraviton gas states. In the black-hole regime of parameters, one desires more entropy than is provided by the supergraviton gas. There are other methods of incorporating D0-brane charge in this regime, which often go beyond supergravity. For example, one can place branes in the type IIA background that carry D0 charge as a worldvolume ux. One possibility is to add a D4-D2-D2-D0 center, which uplifts in M-theory to a M5-M2-M2-P supertube that rotates along the AdS3 [27]. Since supertubes can have arbitrary shapes [28], the solutions corresponding to these con gurations can have a non-trivial dependence on the M-theory circle; however, like the pure D0-brane sources, these are again naively singular con gurations. Estimates of the entropy of back-reacted solutions thus far yield results sub-leading relative to the black hole entropy [27]. Another possibility is to add D0 branes via world-volume ux on a dipolar, egg-shaped D2-brane [22]. Counting the Landau levels of this two-brane has been argued to reproduce the BPS black hole entropy. This \egg-brane" uplifts in M theory to an M2 brane wrapping { 3 { the S2 and spinning in AdS3, and the corresponding solution is also singular [29, 30]. Moreover, simply wrapping branes around this S2 adds yet another charge to the system, which either (a) introduces an uncanceled tadpole which changes the asymptotics of the supergravity elds; or (b) breaks all of the supersymmetry [31]. Either way, such con gurations cannot represent BPS microstates of the original black hole. It is possible that smooth geometric oscillations of superstrata in deep scaling geometries might contribute a nite fraction of black hole entropy [32], but this is by no means proven. Such states lie well beyond the consideration of a supergraviton gas in the [global AdS3] S2 background. Our construction gives new possibilities for deep superstrata in the M-theory frame, and thus represent another advance in the quest for a geometrical understanding of black hole entropy. The elds that make smoothness of superstrata geometries possible are exactly of the kind one expects to see when one considers the back-reaction of the momentum-carrying M5 brane source in the MSW system, or that are generated in string emission calculations [33] in a U-dual four-charge con guration of D3 branes [34].5 We believe that this is not a coincidence but rather an indication that our construction is closing in on a good holographic description of the microstates of this system. Besides its interest for understanding the MSW CFT, our map is also a very powerful solution-generating device. Indeed, we will use it to construct new smooth solutions of ve-dimensional ungauged supergravity that are, in principle, parameterized by arbitrary functions of at least one variable.6 There is a long history of constructing smooth solutions in this theory [20, 21, 35]. However, while these solutions have non-trivial topology, they also have much symmetry; the solution spaces depend on several continuous parameters that describe the location of the topological bubbles. Until now it was not known how to construct smooth solutions in these theories parameterized by arbitrary continuous functions | such solutions were believed to exist only in supergravity theories in space-time dimensions greater than or equal to six, such as those in [4, 11, 13, 14, 36]. Our map thus, in principle, yields the largest family, to date, of smooth solutions of ve-dimensional ungauged supergravity. It also establishes that ve-dimensional supergravity can capture smooth, horizonless solutions with black-hole charges, to a much greater extent than previously thought. The structure of the presentation is as follows. In section 2 we introduce the class of six-dimensional BPS D1-D5-P geometries of interest, and the BPS equations that they satisfy. torus We work with asymptotically AdS3 S3 geometries that can be written as a bration, with the ber coordinates (v; ) asymptotically identi ed with (roughly speaking) the AdS3 angular coordinate and the Hopf ber coordinate of S3 respectively. We introduce maps that involve an SL(2; Q), action on the torus ber and a rede nition of the periodicities of these coordinates, and call these maps \spectral transformations". In section 3 we illustrate the action of a particular spectral transformation on the example of the round, -wound multi-wound supertube solution. This transformation introduces KK 5These elds are absent in the solutions of [24, 27, 29, 30]. 6For our explicit example solutions we will restrict attention to a sub-class of solutions parameterized by one integer, however by the above discussion, the broader family of these solutions is in principle described by arbitrary functions of at least one variable. { 4 { monopole charge into the D1-D5 system. We then recall the known U-duality that relates the D1-D5-KKM system to the MSW system on T6 (or T2 K3). In section 4 we derive the e ect of general SL(2; Q) transformations on the six-dimensional metric and gauge elds. In section 5 we apply our particular transformation to the D1-D5-P superstrata of [11, 14, 37], and we work out an explicit example in detail in section 6. In section 7 we investigate the question of whether there is a weakly coupled symmetric orbifold CFT in the moduli space of the MSW system, as there is for D1-D5. When the compacti cation manifold, M, is T4, the energetics of U(1) charged excitations can be inferred from a supergravity analysis [17], and places strong constraints on the CFT, leading to a no-go theorem. In section 8 we discuss our results, and the appendices contain various technical details. 2 2.1 D1-D5-P BPS solutions and spectral transformations D1-D5-P BPS solutions In the D1-D5-P frame, we work in type IIB string theory on M4;1 S 1 M is either T4 or K3. We shall 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, M, where y y + 2 Ry : We reduce on M and work in the low-energy supergravity limit. That is, we work with six-dimensional, N = 1 supergravity coupled to two (anti-self-dual) tensor multiplets. This theory contains all the elds expected from D1-D5-P string world-sheet calculations [33]. The system of equations describing all 18 -BPS, D1-D5-P solutions of this theory was found in [38]; it is a generalization of the system discussed in [39, 40] and greatly simpli ed in [41]. For supersymmetric solutions, the metric on M takes the local form: coordinate, y, and a time coordinate t via 1 2 u = p (t y) ; v = p (t + y) : 1 2 However, there is some freedom in choosing such a relation, since the form of the metric (and the ansatz in general) is invariant under the shift u0 u 1 2 c0v ; F 0 F + c0 ; !0 ! 1 2 c0 : Using this freedom, we will shortly choose a di erent relation between u, v, t and y that is more natural for spectral transformations and for reduction to ve dimensions. While all the ansatz quantities may in principle depend upon v, throughout this paper we shall require the metric, ds24(B), on the four-dimensional spatial base, and the bration { 5 { (2.1) (2.2) (2.3) (2.4) vector, , to be independent of v. This greatly simpli es the BPS equations and, in particular, requires that the base metric be hyper-Kahler and that d be self-dual on B. The metric and tensor gauge elds are determined as follows. We introduce an index I = 1; : : : ; 4, and an index a that excludes I = 3 (which plays a preferred role): a = 1; 2; 4. The ansatz then contains four functions Za and F , and four self-dual 2-forms, 1; : : : ; 4. These can depend both upon the base, B, and upon the v ber. The function, F , (I), I = appears directly in (2.2) and the warp factor, P, in the metric is given by The vector eld, , de nes (3): (3) P = Z1 Z2 Z42 : d ; The individual functions, Za, and the remaining 2-forms, magnetic components of the tensor gauge elds. (a), encode the electric and Recall that the N = 1 supergravity multiplet contains a self-dual tensor gauge eld, so that adding two anti-self-dual tensor multiplets means that the theory contains three tensor gauge elds. Roughly speaking, the pairs (Z1; (2)) and (Z2; (1)) describe the elds sourced by the D1 and D5 brane distributions. The function, F , and the vector eld, , encode the details of the third momentum charge. In the IIB description, the addition of (Z4; (4)) allows for a non-trivial NS-NS B- eld as well as a linear combination of the R-R axion and four-form potential with all legs in the internal space M; these elds arise in D1-D5-P string world-sheet calculations [33], so are expected to be generically present. For more details, see [38]. The remaining simpli ed BPS equations come in two layers of linear equations. To write them, we denote by d(4) the exterior derivative on the four-dimensional base, and we de ne the operator, D, acting on a p-form with legs on the four-dimensional base (and possibly depending on v), by: { 6 { D d(4) The rst layer of equations determines the Maxwell data. For notational convenience throughout the paper, we work with a form of the BPS equations that is not explicitly covariant in the indices a = 1; 2; 4. In particular, we will always label index, while continuing to refer to these quantities collectively as 4 with a downstairs (a); hopefully this will not cause confusion. The covariant form of the BPS equations is given in appendix A.7 In our conventions, the rst layer of the BPS equations takes the form 4DZ_1 =D (2) ; 4DZ_2 =D (1) ; 4DZ_4 =D 4 ; D 4 DZ1 = D 4 DZ2 = D 4 DZ4 = (2) ^ d ; (1) ^ d ; 4 ^ d ; (2) = 4 (2) ; (1) = 4 (1) ; 4 = 4 4 : 7To pass to the covariant form, one rescales (Z4; 4; G4) ! (Z4; 4; G4)=p2; more details are given in appendix A. The second layer of equations determines the other parts of the metric in terms of the Maxwell data: 2 4 Z2) 4 (1) ^ (Z_1Z_2 (Z_4)2) (2) 4 ^ the following simple parametrization of solutions to (2.6): = K3 V (d ponents parallel and perpendicular to the - ber: ! = (d + A) + $ : We record here the ansatz for the three-form eld strengths in terms of the above data. A discussion of how these eld strengths appear in the corresponding Type IIB ansatz may be found in [38] and a simpli ed version without (Z4; 4) may be found in [41]. The BPS ansatz for the uxes, where ds24 and are v-independent, is given by:8 G(1) = d G(2) = d G4 = d These elds satisfy a twisted self-duality condition; since this is most conveniently expressed in covariant form [43] (see also [44]), we give it in appendix A. 2.2 Canonical transformations As noted above around eq. (2.4), there is some freedom in relating the (u; v) coordinates to the time and spatial coordinates, (t; y). As will shortly become clear, it will be convenient for us to use the following relation throughout this paper: u = t ; v = t + y ; y = y + 2 Ry : (2.14) 8Note that, following [42], we have rescaled (1;2) ! 21 (1;2) relative to the conventions of [41]. See also Footnote 7. { 7 { (2.9) (2.11) (2.12) 2 One should note that the coordinates (t; y) are the same as those in eq. (2.3). To get from eq. (2.3) to the above relation, one can make a shift (2.4) with c0 = rescaling of u0 by p1 and v by p2, together with accompanying rescalings of the ansatz 2, followed by a quantities, as follows. First, we take c0 = 2 in (2.4) and de ne: u0 u + v ; F 0 F 2 ; !0 ! + One then arrives at (2.14) by dropping all the tildes. Most importantly, with these re-scalings, the ansatz for the metric, the ansatz for the uxes and the BPS equations remain unchanged: the factors of p 2 cancel throughout. Thus we are free to use either coordinate representation, (2.3) or (2.14), solve the BPS equations and substitute into the ansatze: both will produce BPS solutions. The resulting solutions will of course be related by (2.15) and (2.16). We illustrate this point with a simple supertube solution in appendix B. The u = t parameterization is much more convenient when comparing six-dimensional solutions to ve-dimensional solutions. Assuming that F is everywhere negative, as it will be in our solutions, one can write the metric as: 1 P F ds62 = p (du + !)2 F p P dv + + F 1(du + !) 2 + p Since F is everywhere negative, the v coordinate is everywhere spacelike. Solutions with an isometry along v can be reduced on v to obtain ve-dimensional solutions. In this reduction, u is the natural time coordinate in ve dimensions. This is the advantage of the Finally, to be clear: in this paper we will use (2.14) and u = t will be kept xed in all u = t parameterization for our purposes. spectral transformations. 2.3 General spectral transformations Note that for solutions with a GH base, the six-dimensional solution in the form (2.17), (2.10) is written as a double circle bration, de ned by (v; ), over the R base of the GH metric. In this paper we will exploit a set of maps that involve coordinate 3 transformations of the (v; ) coordinates. We consider maps that act on (v; ) with elements of SL(2; Q) and not just SL(2; Z), and so in general one must be careful to specify how the map acts on the lattice of periodic identi cations of these coordinates. Our maps consist of a composition of a coordinate transformation and an accompanying rede nition of the lattice of coordinate identi cations. The coordinate transformation component of our map is an SL(2; Q) map that transforms a solution written in terms of (v; ) coordinates to a solution written in terms of { 8 { new coordinates (v^; ^). We parameterize the SL(2; Q) action by rational numbers a; b; c and d subject to ad bc = 1, as follows: v R = a v^ R For later convenience, in the above we have introduced the shorthand R for the ratio of the periodicities of the v and coordinates, so that the linear transformation acts on circles with the same period of 4 . We emphasize again that the coordinate u is held xed. The lattice rede nition component of our map is as follows. We consider starting con gurations for which the lattice of identi cations is9 We de ne the new lattice of identi cations of the new solution to be v = v + 2 Ry ; = + 4 : that is, the new lattice is not the one that would follow from making the coordinate transformation (2.18) on the original lattice (2.19), but is rede ned to be (2.20). The fact that the lattice is rede ned means that when the parameters of the map are non-integer, the maps are in general not di eomorphisms and can modify the presence or absence of orbifold singularities in the spacetime, as has been observed in fractional spectral ow transformations [10]. We illustrate the above procedure by reviewing the example of fractional spectral ow transformations of multi-wound circular D1-D5 supertubes in appendix C. We will refer to these maps as \spectral transformations". Having made the above transformation, one can recast the metric and tensor gauge elds back into their BPS form but in terms of the new coordinates, (v^; ^). For example, one substitutes the coordinate change (2.18) into the metric (2.17) and (2.10), and then rewrites the result as: 1 p Pb Fb + p This rearrangement of the background metric and tensor gauge elds in terms of the new bers de nes new `hatted' functions and di erential forms in terms of the old functions and forms. We will derive the explicit transformation rules for the individual ansatz quantities in section 4, and use these rules to transform the family of superstrata solutions that we consider. After this transformation the local metric is still the same as the original one, so the background is still locally supersymmetric, and so the hatted ansatz quantities solve the BPS equations in the form (2.8), (2.9). More speci cally, if the original functions and forms 9For ease of exposition, here we suppress possible additional identi cations that involve and an angle in the three-dimensional base; we will be more precise when we discuss explicit examples later. { 9 { in the solution depend upon (v; ) then that dependence must, of course, be transformed to (v^; ^) using (2.18), and the BPS equations satis ed by the hatted quantities will be those of (2.8) and (2.9) but with (v; ) replaced by (v^; ^). While the transformed solution is still locally supersymmetric, it is possible that the rede ned lattice of identi cations may break some supersymmetry; indeed, we shall see that the transformation that we employ in the current work will break half of the eight real supersymmetries preserved by the D1-D5 circular supertube solution. Five-dimensional solutions and spectral transformations Solutions that are independent of v can be dimensionally reduced from six to ve dimensions. The BPS equations become those of N = 2 supergravity coupled to three vector multiplets. In particular, the description of the four vector elds of this theory involves totally symmetric structure constants, CIJK . Indeed, for the system we are considering one has with all other independent components equal to zero.10 written as follows [35]:11 The complete family of smooth solutions that are also -independent may then be (2.22) (2.23) (2.24) (2.25) (2.26) (I) = dBI ; ZI = LI + CIJK 1 2 = M 2 + KI LI + We now reduce the metric, (2.17), on the v- ber. Following (2.14), we set u = t, and we relabel ! and F in terms of their more standard ve-dimensional analogs: k ! ; Z3 F : This yields the standard ve-dimensional metric: 10One can convert this to the canonical normalization (in which C344 = 1) by the procedure described in Footnote 7. 11Note that our convention for M di ers from that of [35] by a factor of 2. Spectral transformations for v-independent solutions The role of SL(2; Z) spectral transformations on (v; ) was studied in detail for vindependent solutions in [45]. In particular, the spectral transformations could be reduced to transformations on the harmonic functions V; KI ; LI and M . Moreover, from the vedimensional perspective, any of the Maxwell elds can be promoted to the Kaluza-Klein eld of the six-dimensional formulation and so there as many di erent SL(2; Z) spectral transformations as there are vector elds. Moreover, these SL(2; Z) actions do not commute and, in fact, generate some even larger sub-group of the U-duality group. The original study of spectral transformations was made for the system with two vector 0) but the results can be recast in a form that is valid for ve-dimensional N = 2 supergravity coupled to (NV 1) vector multiplets, and so we will give the relevant general results. The spectral transformations considered in [45] included two important sub-classes: \gauge transformations" and \generalized spectral ows". A gauge transformation is generated by choosing one of the Maxwell elds as the KK eld, then leaving xed and shifting v by a multiple of . The choices of uplift lead to NV gauge parameters, gI , and the gauge transformations reshu e the harmonic functions according to: Vb = V ; LbI = LI Mb = M b KI = KI + gI V ; CIJK gJ KK gI LI + 1 2 CIJK gJ gK V ; 1 1 2 CIJK gI gJ KK + 3! CIJK gI gJ gK V : While this is a highly non-trivial action on the harmonic functions, this transformation leaves the physical elds, ZI ; (I), transformations. and $ invariant, and hence their designation as gauge Spectral ows are induced by keeping v xed and shifting by a multiple of v. Again there are NV ways to do this with NV parameters, I , resulting in the full family of generalized spectral ow transformations. In terms of CIJK de ned as CIJK II0 JJ0 KK0 CI0J0K0 ; generalized spectral ow transformations act on the harmonic functions as follows: Mb = M ; b KI = KI Vb = V + I KI LbI = LI I M ; CIJK J LK + 2 1 CIJK J K M ; 2 1 CIJK I J LK + 1 3! In contrast to the gauge transformations, these transformations have a complicated and very non-trivial action on the ve-dimensional physical elds (see [45]). In our conventions, the polarization direction I = 3 in eq. (2.29) corresponds to the (Kaluza-Klein) vector eld in ve dimensions that lifts to metric in six dimensions. We reserve the term \spectral ow" for generalized spectral ows in this polarization direction. (2.27) (2.28) (2.29) Spectral ow transformations have the same e ect as the following large coordinate transformation, where the new coordinates are denoted with a hat: and where the other coordinates are invariant. In the D1-D5 system CFT, the world volume of the CFT lies along the v- ber and translations along this ber are generated by the Hamiltonian, L0. The - ber lies transverse to the D1 and D5 branes and so represents an R-symmetry transformation. The above transformation is thus a CFT spectral ow; for a more detailed discussion, see [5, 6, 10]. Similarly, a gauge transformation in the polarization direction 3 with parameter g3 has the same e ect as the following coordinate transformation, where again the new coordinates are denoted with a hat: v = v^ + g3 ^ ; = ^ ; (2.30) (2.31) and where the other coordinates are invariant. Note that, when the world-volume of the CFT lies along the v- ber, this seemingly trivial (from a supergravity point of view) transformation does not appear to have a simple interpretation in the dual CFT. The transformation would appear to reorient the world-volume of the CFT, and the question of whether there is any sensible holographic interpretation of this gravity transformation remains somewhat mysterious. However, if one interchanges the roles of and v such that the world-volume of the CFT lies along the - ber, and the Hopf ber of the S3 lies along v, then (in our conventions) the above gauge transformation would correspond to spectral ow in the left-moving sector of the CFT. 3 The multi-wound supertube and mapping D1-D5 to MSW Before proceeding to general spectral transformations, it is very instructive to see how spectral transformations act on one of the most important v-independent BPS solutions: the multi-wound supertube [18, 19, 28, 46]. We start with its standard formulation as the smooth geometry of the D1-D5 supertube, and we map it to M-theory with an SL(2; Q) spectral transformation and a U-duality transformation. The SL(2; Q) spectral transformation introduces a KKM charge along the Hopf ber of the S3, and the D1, D5 and KKM charges transform under the U-duality into three independent M5-brane charges underlying an MSW string, where the resulting con guration is a particular form of that string, with speci c dissolved M2-brane charges and speci c angular momenta. There is the following interesting conundrum: the D1-D5 supertube is 14 -BPS, preserving eight supersymmetries, while the MSW string is 18 -BPS and preserves only four supersymmetries. We reconcile this di erence by carefully examining the lattice of identications, and showing that our transformation accounts for the change in the number of supersymmetries. 3.1 take F = with (2.14) is given in appendix B. The canonical starting point for the multi-wound, circular D1-D5 supertube is the ansatz (2.2) with a time coordinate, t, and the asymptotic S1 coordinate, y, related to the coordinates u; v via (2.3) and with F = 0. However, as we stipulated earlier, we are going to use (2.14), and the transformations (2.15) and (2.16) then imply that one must 1. The precise relation between the supertube with (2.3) and the supertube The -wound supertube is a two-centered con guration de ned by the following harmonic functions: V = L1 = 1 r+ ; Q1 ; The base metric is at R4, which we write as The remaining ansatz quantities for the supertube are then: Z1 = = Q1 ; Rya2 The parameters are subject to the following regularity condition: Q1Q5 = 2Ry2 a2 : This solution may be written in Gibbons-Hawking form by de ning new coordinates, ( ; ; # ) via: One then has 1 r + A)2 + V hdr+2 + r+2 (d#2+ + sin2 #+ d 2)i ; V = 1 r+ = 4 ; A = cos #+ d = (r2 + a2) sin2 r2 cos2 (d'1 (3.1) (3.2) (3.3) (3.4) (3.5) (3.6) (3.7) measure the distances in the at three-dimensional base between two centers, de ned by r = 0; one can choose Cartesian coordinates in which we have r = This solution describes a -wound supertube, whose KKM dipole moment is . Given the above choice of gauge for the one-form A, the lattice of identi cations for this solution is HJEP06(217)3 generated by12 (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) (y; ; ) (y; ; ) (y; ; ) (y + 2 Ry; ; ) ; (y; (y; + 4 ; ) ; + 2 ; + 2 ) : There is a Z orbifold singularity at the supertube locus, as we will now review. Introducing the coordinates r=a sinh ; = 2 ; y~ = y Ry ; t~ = the metric can be written as ; 2 Under the further change of coordinates we observe that the metric is locally AdS3 S3, patches around the North and South Poles of the S2 respectively. The identi cations on 0 and are then simply 0 = 0 + 4 at xed , and = + 2 at xed y~; ~; ~ y~; ~; ~ y~; ~; ~ y~ + 2 ; ~ 2 ; ~ 2 ; y~; ~ + 4 ; ~ ; y~; ~ + 2 ; ~ + 2 = 1, the full geometry is simply global AdS3 S3, and when identi cation above is an orbifold identi cation that combines the AdS3 and the S3, and that gives rise to a Z orbifold singularity at the location of the supertube, (r = 0; = Spectral transformations of the supertube We now perform an SL(2; Q) map on the above multi-wound supertube con guration, that maps the solution to a form in which it can be straightforwardly dualized to the MSW frame. Our SL(2; Q) map can be decomposed into a product of gauge and spectral ow transformations, and it will be instructive to go through these steps. We rst perform a gauge transformation with parameters (g1; g2; g3) = (0; 0; 12 Ry). The resulting harmonic functions are: V = L1 = ; 1 K1 = K2 = 0 ; K3 = L2 = Q5 ; 4 r Ry 1 2 r ; We next perform a (fractional) spectral ow transformation with parameters ( 1; 2; 3) = (0; 0; 2=( Ry)). The resulting harmonic functions are: V = L1 = 1 ; K1 = L2 = Q5 1 2 Ry r Q5 ; 4 r ; K2 = L3 = Q1 1 2 Ry r Q1Q5 4 2Ry2 r ; ; K3 = M = Ry 1 2 r ; Ry + 2 = ( 4QR5y ; 4QR1y ; R4y ). The resulting harmonic functions are (here I = 1; 2; 3 and we employ notation mod 3 for the I indices): where we have de ned V = LI = 1 r+ ; k 2 KI = M = Q1 ; 2 Ry 1 r+ k I 2 k k k + Ry 2 ; 1 r : 2c : (3.18) (3.19) These harmonic functions are those that describe the MSW maximally-charged Ramond ground state solution in ve dimensions (related by right-moving spectral ow to the NS vacuum) [22, 47], as reviewed in appendix D. We will review the duality map momentarily. The combination of these transformations corresponds to the following SL(2; Q) map on the coordinates: v Ry = and where as discussed above, we rede ne the new lattice of identi cations. The new lattice is generated by the appropriate smooth identi cations in the M-theory frame; combining (2.20) with the appropriate smooth identi cation on as discussed around (D.10), the new identi cations are: where for each of the three generators in this equation, one holds the other two periodic coordinates in the equation xed. becomes Note that in the tilded coordinates, de ned in (3.13), the coordinate transformation v Ry = ^ 2 ; ~ = v^ k3 ; (u; ) xed ; which can be described as a (fractional) \spectral interchange" transformation [42] between rotating versions of the AdS3 circle coordinate and the Hopf ber coordinate of the S3. Indeed, under this transformation, the metric (3.12) transforms to: (3.21) (3.22) (3.23) (3.24) (3.25) cosh2 dt~2 + d 2 + sinh2 d'2 where we de ne + 1 4 dv^ k3 + cos d ~ 2 It is interesting to re-interpret the coordinate transformation (3.22) in terms of the D1-D5 and MSW CFTs. The relation between ~ and in (3.13) corresponds to spectral ow in the left-moving sector of the D1-D5 CFT; the solution in terms of corresponds to a particular Ramond ground state, while the solution in terms of ~ corresponds to the NS vacuum of the left-moving sector. (The analogous statement holds for ~ and in terms of spectral ow in the right-moving sector of the CFT.) The coordinate transformation in the tilded coordinates (3.22) is interesting, as we see from it that v^ is a rescaled version of ~, the left-moving NS sector coordinate. One can paraphrase these observations by describing the metric (3.23) as being written in NS-NS sector coordinates, corresponding to the NS-NS vacuum of the dual CFT state, which has L0 = L0 = 0 : Qe1 = Q1 Qe5 = Q5 : Qe1 = gs 03n1 V4 ; Qe5 = gs 0n5 ; n1 = N1 n5 = N5 : Recalling the usual relation between Qe1;5 and n1;5: we see that the relation between the integer brane numbers on the two sides of the map is If one rewrites ~ in terms of , one can describe the metric as being expressed in NS-R sector coordinates, and corresponding to the NS-R ground state obtained from the NS-NS vacuum via right-moving spectral ow with parameter 1/2, which has ber, and these quantum numbers will correspond respectively to the NS vacuum and the maximally-charged R ground state of the MSW CFT (which we again emphasize are related by right-moving spectral ow). Let us analyze the lattice of identi cations (3.21) that has resulted from our transformation. The AdS3 angle coordinate ' has period 2 , which is the correct periodicity for a smooth global AdS3. The combination that appears in place of the Hopf ber of the S3 is v^=k3, which has period 4 = , corresponding to a smooth Z quotient of the Hopf ber, appropriate for the decoupling limit of a D1-D5-KKM con guration with KKM charge . Note now that the relation between the dimensionful parameters Q1, Q5 and the integer number of D1 and D5 branes n1, n5 that correspond to this solution is changed, relative to the solution without KKM charge. The Gaussian integral de ning the charges is now done over a range of the Hopf ber coordinate that is smaller by a factor of , so that the actual new supergravity charges are ; ; (3.26) (3.27) (3.28) (3.29) (3.30) We note that using (3.27), the relation (3.19) becomes k1 = Qe5 ; 2Ry k2 = Qe1 ; 2Ry k3 = Ry 2 : so that, up to constant factors, the parameters kI correspond to the D1, D5 and KKM charges, which map to the three di erent M5 charges in the M5-M5-M5 duality frame. The other e ect of the rede ned lattice of identi cations is that, in the asymptotically AdS3 S3 geometry, it breaks the SU(2)L SU(2)R symmetry of the S3 down to U(1)L SU(2)R. Since this is the R-symmetry, it must break the N = 4 superalgebra of the left-moving sector down to an N = 2 superalgebra with this U(1)L R-symmetry. Since these remaining supercharges are charged under the translations along the Hopf ber, they will not survive the dualization to M-theory and thus the e ect of reassigning the lattice identi cations and compactifying is to break all the left-moving supersymmetries even in the ground-state con guration we are studying here. so we obtain with The duality map from D1-D5-KKM to M-theory involves T-duality on the Hopf ber of the S3 and two directions in the T4, followed by an M-theory lift.13 This results in a solution which asymptotically has a compact T6. The e ect of these dualities from the point of view of the lower-dimensional theory is encoded in a dimensional reduction on the Hopf ber of the S3, to ve dimensions (see e.g. [51, 52]). The SL(2; Q) transformation we have performed means that the ansatz quantities have already been rearranged to make this step straightforward: the coordinate v^ is precisely the Hopf ber of the S3. The metric (3.23) can be written (using an obvious shorthand) as ds62 = pQ1Q5 ds2AdS3 + pQ1Q5 4(k3)2 dv^ + k3 cos d ~ 2 + 4 pQ1Q5 ds2S2 : The reduction ansatz for the six-dimensional metric takes the form ds62 e 3A(dv^ + A^(3))2 + eA ds52 ; Using the relation (3.19), and as reviewed in appendix D, this becomes exactly the decoupled M5-M5-M5 metric in ve dimensions that results from the set of harmonic functions (3.18) [22] (see also [27, 47, 53]): Note that the smooth Z quotient on the Hopf ber has migrated into an M5 charge, and thus a parameter in the warp factors, and the ve-dimensional solution is smooth AdS3 S with standard coordinate identi cations. 4 General SL(2; Q) transformations in six dimensions Having understood how to map the ground-state of the D1-D5 system onto that of the MSW string, we now wish to extend our results to the transformation to families of leftmoving excitations. This includes spectral transformations of superstrata [11, 13, 14]. We start more generally by considering a generic BPS background that can depend on all of the coordinates, except, of course, u. We recall our parameterization of the general spectral transformations on (v; ) from eq. (2.18): 13For early works on reduction and T-duality along Hopf bers, see [48{50]. v^ R = a where The expression for ^(2) then simpli es signi cantly: where in turn d^(4) is the exterior derivative on the transformed four-dimensional base. The remaining uxes b (1) and b 4 may be obtained similarly. To reduce to ve dimensions we require solutions that are independent of the v^- ber. This means restricting the modes to those with k = 2m, and hence with a phase, ^k;m;n, in (5.14) given by: ^2m;m;n = (n + m) ^ m : (2) satisfy the rst layer of the equations: One can then nd b (2) using (4.19), whereupon one can explicitly verify that Zb1 and Db b (2) = 0 ; Db ^4Db Zb1 = b (2) ^ d b ; b (2) = ^4 b (2) ; D b d^(4) ; (5.28) (5.29) (5.30) (5.32) 1 2 a2(m + n) 4 Ry ^(2) = 1 : (5.31) The BPS equations also reduce to their ve-dimensional form and, in particular, (5.28) implies that b (2), is self-dual and closed: d^b (2) = 0 ; b (2) = ^4 b (2) ; and is thus \harmonic" on the GH base. One can explicitly verify this using (5.31) and (4.19). The harmonic forms on a standard Riemannian GH base are well-known (see, for example, [35]). For NC GH centers there are (NC 1) independent, smooth harmonic forms given by the expressions in (2.23). In particular, these harmonic forms are independent of the angles ( ; ). It may therefore seem surprising that there is, in fact, a doubly in nite family of \harmonic forms" emerging from our solutions. However, this is because the base is ambi-polar and hence singular on the locus Vb = 0. This singular locus enables the \harmonic forms" to have (singular) sources on this locus and thus the system admits large families of solutions with oscillating magnetic uxes. Again, as with everything else in the ambi-polar formulation, the physical eld strengths must be smooth. In this paper smoothness is guaranteed because we derived the solution by a coordinate change of a smooth six-dimensional solution. Henceforth we will use the term pseudo-harmonic forms to refer to the generalized \harmonic forms" that are singular on the degeneration locus (Vb = 0) of an ambi-polar geometry, and yet give rise to smooth physical elds in the complete solution. The rst analyses of ve-dimensional BPS solutions were done over a decade ago [20, 21, 35] and the pseudo-harmonic forms were missed in that analysis. Given the ambi-polar structure of the base, many people were aware that the singular locus could allow the presence of new sources that could generalize the usual known solutions. The problem was that there was a vast range of singular sources available and no obvious systematic way to nd precisely those sources that would lead to smooth physical eld strengths. That is, the possibility, let alone the classi cation, of non-trivial pseudo-harmonic forms remained unclear. It is interesting to note that the possibility of ambi-polar metrics was rst found by Giusto and Mathur [55] by studying spectral ows of smooth supertube geometries. In this paper we have used more general spectral transformations to discover precisely how to go beyond the standard analogs of Riemannian harmonic forms in ve dimensions to obtain (hopefully complete15) families of pseudo-harmonic forms on our speci c ambipolar geometry. It would be very interesting to see how pseudo-harmonic forms might be characterized, in terms of the di erential geometry, and then computed for generic ambi-polar hyper-Kahler metrics. The bottom line is that we have obtained a huge class of pseudo-harmonic forms and these lead to new families of smooth ve-dimensional solutions with uctuating uxes. As we have argued above, these solutions must be dual to microstates of the MSW string. 6 An explicit example We now give a complete explicit example. It is one of the family of solutions discussed in the previous section and has parameters (k; m; n) = (2; 1; n). Since k = 2m, this can be dualized to a smooth ve-dimensional solution. 6.1 The D1-D5-P superstrata The quantities and ds24 are again as given in eqs. (3.4) and (2.10). The quantities of the rst layer of the BPS equations are as given in (5.7) with (k; m; n) = (2; 1; n), for a non-negative integer, n, where (n + 1) is a multiple of . The phase dependence of this solution is: 2;1;n = (n + 1) v Ry ) ^2;1;n = (n + 1) ^ : 1 2 The relation between b and b4 required for regularity is b2 = b 2 4 Again generalizing the solution of [14] to > 1, the solution to the second layer of BPS equations is: F = 1 ! = !1 d'1 + !2 d'2 ; b 2 2 a2 + b 2 4 2 a2 sin2 4;2;2n cos2 4 a2 + 1 ; (6.1) (6.2) (6.3) Speci cally, while singular on the degeneration locus of ambi-polar geometries, pseudo-harmonic forms are required to lead to smooth BPS solutions in ve dimensions. !2 = b42 Ry 4;2;2n a2 1 + b 4 1 4;2;2n r2 + a2 1 + r 2 (n + 1) a2 cos2 We record here the values of the other ansatz quantities that will be used when mapping to the M-theory frame: P = Q1Q5 2 Ry 2 1 a2 + a2 + 2 2 2 2 Ry r2 4;2;2n ; 24 Ry 2 sin2 cos2 + b24 Ry r2 4 4;2;2n r2 + a2 a2 1 + 1 These quantities lead to a family of smooth, CTC-free solutions, due to the coi uring ansatz and appropriate choices of homogeneous solutions to the BPS equations [14]. To transform to the M-theory frame we convert the base metric to GH form and transform the ansatz quantities recorded above. Using (4.11) and (4.2), the metric functions in the M-theory frame are Zb1 Zb2 = Pb = Q1 Q5 Q1Q5 ; )2 + Z(osc) = 1 Q1 Zb4 = b 2 4 1 2 a2 + b2 4;2;2n : 1 + b 2 4 2 a2 + b2 b4 Ry 2;1;n cos ^2;1;n ; 4;2;2n cos ^4;2;2n ; (6.5) The one-forms, ^(I), are obtained from (4.18) with c = 1 , ^(1) = 1 R D (Vb 1Z2) ; 1 R (2) D (Vb 1Z1) ; ^ 4 = 4 where we have used that (1) = 0 from (5.7). We nd 1 R D (Vb 1Z4) (6.6) ^(1) = ( + ) Zb2 ; 4 Ry ( + ) Zb1 4 Ry ( + ) Zb4 4 Ry + + 4 Ry 4 Ry (n + 1) a2 h4 cot 2 Zb1(osc) d (n + 1) a2 h2 cot 2 Zb4 d + i ; (6.7) + i where we recall our notation that d(3) is the exterior derivative on the R 3 base of the GH space. From the above expressions one obtains the b (a) using (4.19). These are manifestly self-dual, and it is straightforward to verify that they are indeed closed. The transformations that yield the last layer of the BPS system are (4.6), (4.9) and (4.10). Using these with c = 1 , we obtain: Fb = ^ = 1 Ry 2( 2 2 2 2 2 4 4 4;2;2n ; cos2 4;2;2n a2 sin 2 2r2(r2 + a2) 1 + n cot 2 a2(2r2 + a2) tan 2 ; (6.8) One can then verify that these quantities, together with the hatted quantities and ansatz given in (5.15){(5.20), do indeed satisfy the last layer of BPS equations for k = 2 and m = 1. Smoothness in ve dimensions requires that ^ and the ZbI are nite at the GH points while the absence of CTC's requires that ^ vanishes at the GH points. The GH points lie at (r; ) = (0; 0) and (r; ) = (0; =2) and if one sets r = 0 in (6.5) and (6.8), one has: Zb1 = Zb3 = Q1 a2 cos 2 Fb = ; 1 a2 cos 2 Zb2 = a2 + a2 cos 2 Q5 b 2 2 ; Zb4 = 0 ; ; ^ = a2 + tan2 2 : Ry 4 a2 b 2 2 (6.9) A complete analysis of the global absence of CTCs is in general a di cult problem, often relying on numerical tests, and is beyond the scope of this paper. Here we content ourselves with observing that the ve-dimensional solution satis es the requisite local conditions, providing evidence that the spectral transformation indeed maps the CTC-free D1-D5-P superstratum onto a CTC-free solution in the M-theory frame. 7 Comments on symmetric product orbifold CFTs It is a tantalizing prospect that the D1-D5-KKM system might have a solvable CFT in its moduli space, given that it is so similar to the D1-D5 system | di ering only by a discrete identi cation on the transverse angular S3. One might think that since, in the decoupling limit, the introduction of KKM charge to the D1-D5 geometry amounts to a Znk orbifold of the Hopf ber of S3, that a similar quotient of the dual CFT by a chiral R-symmetry rotation would yield the corresponding dual CFT for the D1-D5-KKM system [56].16 The rst part of the construction in this paper maps a multi-wound D1-D5 supertube to a D1-D5-KKM bound state. It is tempting to translate this into a map between states of the D1-D5 symmetric product orbifold CFT and the putative D1-D5-KKM symmetric product 16For related work on the microstates of the D1-D5-KKM system, see for example [57{63]. orbifold CFT. The multi-wound D1-D5 supertube con guration described in section 3 corresponds to a R-R ground state of the D1-D5 CFT with N1N5= strands each of winding , with the same R-R ground state on each strand. If there existed a symmetric product D1-D5-KKM CFT, for n1 D1-branes and n5 D5-branes and KKM charge nk = , then this CFT should have total number of strands n1n5 = N1 N5 (7.1) where we have used the relation between the brane numbers on the two sides of our map, given in equation (3.29). The map appears to conserve the total number of strands, while mapping strands of winding in the D1-D5 CFT to strands of winding 1 in the D1-D5HJEP06(217)3 KKM CFT. However, when M = T4, strong constraints arise from the structure of U(1) currents and the energetics of states carrying the corresponding charges [16, 17], as we now review. Review of the D1-D5 CFT To begin, consider type IIB supergravity compacti ed on T5. The moduli space of this theory is the 42-dimensional nE6(6)=USp(8), where the U-duality group is E6(6)(Z). Wrapped branes and momentum excitations transform as a 27 under this group; the presence of the background D1-D5 charge vector ~q reduces the moduli space to the 20dimensional Hq~nSO(5; 4)=(SO(5) SO(4)) through the attractor mechanism [64], and the U-duality group reduces to the subgroup Hq~ SO(5; 4; Z) vector decomposes as that xes ~q. The charge 27 ! (1 9) 16 1 (7.2) where the 1 9 represent the \heavy" charges (branes wrapping the y circle, including the D1-D5 background; the second singlet is the momentum charge along the y circle; and the 16 comprises branes and momentum along the T4 but not along the y circle. Elements of SO(5; 4; Z) not in Hq~ do not preserve the charge vector ~q, instead they act as nite motions on the moduli space Hq~nSO(5; 4)=(SO(5) SO(4)). Such transformations are not symmetries of the CFT, any more than any other nite motion on the moduli space preserves the CFT. What such nite motions do tell us is that, if there is a weakcoupling cusp in the moduli space for a given pair of brane quanta (n1; n5), then there are other cusps in the moduli space where the dual CFT becomes weakly coupled, one for each factorization of N = N1N5 into any other pair of integers (N10 ; N50 ) with N = N10 N50 [16, 65]. Because these are motions on the moduli space and not symmetries, the existence of a locus in the moduli space described by a symmetric product orbifold in one cusp does not imply the existence of such a description in any other cusp. The question then arises, in which cusp does the symmetric product orbifold (T4)N=SN lie? The BPS mass formula for the 27 is on one hand protected by supersymmetry, and on the other hand depends on the moduli and so determines the answer [16]. In the decoupling limit of the D1-D5 system, the energetics of the 16 is (for a rectangular torus with all the antisymmetric tensor moduli switched o ) hR = 1 4 X 4N1 i=1 pg pi r i 1 4 X 4N5 i=1 wF 1 i r gs : (7.3) There is no invariant notion of the \level" of a U(1) current algebra, as the normalization of the current-current two-point function is moduli dependent. For instance, from the previous considerations we know that SO(5; 4; Z) transformations can change the values of n1 and n5 in the above formula. One can however compare the energetics of charged states in the CFT with the above expression. The symmetric product orbifold has four left-moving translation currents (the diagonal sum of the translation currents in each copy of T4), which realize the rst of the two terms in eq. (7.3), if we set N1 = N . The other eight charges can be realized as the winding and momentum charges on a separate copy of T4. The presence of this additional component of the CFT is necessary to realize all the U(1) currents and the wrapped brane charges they couple to. Thus it is natural to associate the symmetric product orbifold with the weak-coupling cusp of the moduli space where the appropriate low-energy description has N1 = N and N5 = 1. The addition of KK-monopole charge compacti es one more dimension of the target space | the bered circle of the KK-monopole (the circle), which is the Hopf in the decoupling limit. One now has type IIB supergravity compacti ed on T6, whose moduli space is E7(7)=SU(8). The charge vector ~q of wrapped branes and momentum on T6 transforms as a 56 of E7(7). The background D1-D5-KKM charges break the moduli space down to the 28-dimensional space Hq~nF4(4)=(SU(2) USp(6)), and the 56 decomposes as ber of S3 56 ! (1 26) (1 26) 1 1 where once again the rst (1 26) is associated to the heavy background of branes wrapping the y circle, and the second such factor is associated to wrapped branes and momentum along the compacti cation S1 T4 transverse to the y circle; the remaining two charges are KK-monopoles whose bered circle is the y circle, and momentum along the y circle. The (1 26) of wrapped branes/momentum charges along S1 T4 are again associated to a set of U(1) currents in the CFT, and once again their energetics can be deduced from the decoupling limit of the BPS mass formula [17] (7.4) (7.5) 2 : hR = pg pi r i i s + wDi1 pg r wD ne + wD~ 1 s e !2 2 1 i r gs d1 n5 + p nk + d5 6789 n1 Here the third octet of charges related to U(1)'s of \level" nk are (f 1 ; n5 6789; d3 ij), and the e are the corresponding volumes of the cycles they wrap, in appropriate units. Once again there is a cusp of the moduli space for every factorization of N into a triplet of background charges n1, n5, and nk; the supergravity description of the CFT is thus merely a low-energy e ective eld theory approximation. This fact also leads to a minor puzzle. The only remnant of KK monopoles in the decoupling limit of the background is a Znk quotient of the angular S3, which breaks the SU(2)L symmetry down to U(1)L SU(2)R, and the supersymmetry from (4; 4) to (0; 4). But when nk = 1, there is no quotient, and so it seems that there is an unbroken (4; 4) supersymmetry. The resolution of this puzzle appears to be that indeed an accidental left-moving N = 4 supersymmetry develops in the decoupling limit, on a codimension 8 sub-locus of the cusp SU(2)R Rof the moduli space corresponding to nk = 1. The moduli space of the D1-D5-KKM system get to any of the other supergravity limits with other values of nk, one must turn on the additional eight moduli that break the accidental N = 4 supersymmetry of the left-movers. Again one can ask whether there is a symmetric product CFT W N =SN somewhere in the moduli space. It is again reasonable to suppose that the component CFT W has four translation currents to generate the winding/momentum contributions in the rst term of equation (7.5). The diagonal current that survives the orbifold projection yields a U(1) of \level" N and so can only match the above energetics in the cusp where one of the background charges is N , and again it is natural to take n5 = nk = 1 and n1 = N . The second and third terms on the r.h.s. are then the contributions of eight more currents of level one, and can be realized with a separate T The last term in the wrapped brane energetics (7.5) is di cult to realize in a symmetric product structure. With n1 = N , n5 = nk = 1, one seeks another current of level N . If the building block is a c = 6 superconformal eld theory on T4 (with once again an extra T 4 T 4 CFT to realize the \level-one" terms), the translation currents comprise c = 4, and their superpartners are four free fermions comprising the remaining c = 2 (at least for the right-moving supersymmetric chirality). Bilinears in the free fermions form a levelone SO(4) = SU(2) SU(2) current algebra, of which one SU(2) is the R-symmetry. The other, \auxiliary" SU(2) has energetics m2=4 for the individual component CFT W, where m is the eigenvalue of J aux for this auxiliary (level-one) SU(2) current algebra.17 3 The symmetric product structure then leads to an energetics m2=4N under the diagonal J3aux. This energetics of SU(2) level-one current algebra is thus incompatible with the last term in equation (7.5) by a factor of 3, and any attempt to engineer the requisite normalization naively leads to a breaking of the (0,4) supersymmetry. One can ask whether this lattice of auxiliary SU(2) charges with energies m2=4N is a sublattice of some larger lattice of CFT zero modes, which also contains the values present in (7.5). The possibilities are constrained by the full structure of U(1) charges in supergravity. The (1 26) charges of wrapped branes/momentum on S1 T4 decompose as (1; 1) (2; 6) (1; 14) (7.6) under the local SU(2) USp(6) symmetry of the moduli space of the D1-D5-KKM background. The thirteen right-moving currents account for (1; 1) (2; 6), with the singlet 17In the symmetric orbifold describing the D1-D5 system, this auxiliary SU(2) is an accidental symmetry of the orbifold locus, and does not survive perturbations away from this locus. associated to the last term in (7.5) and the second factor associated to the translation currents on the various copies of T4; the remaining (1; 14) are related to left-moving currents, for which there is less information due to the lack of supersymmetry in that chirality of the CFT. A reasonable assumption is that twelve of the 14 are the left-moving counterparts of the rst three terms in (7.5) where one ips the relative sign of the \winding" and \momentum" contributions. There are two more special currents whose energetics can then be determined from the local SU(2) USp(6), leading to [17] hL = pg pi r i s 1 4 X 1 2 The spectrum of the one right-moving and two left-moving \special" currents associated to the charges p ; d1 ; d5 6789 in (7.5), (7.7) has also arisen in a related context, in which spectral ows were used to generate a class of nonsupersymmetric solutions.18 Indeed, the charged states are all non-BPS, even though the starting point in the analysis is a BPS mass formula; after the decoupling limit, none of the U(1) currents lie in the stress tensor supermultiplet, even though before the decoupling limit, the right-moving charges did have that property. The U(1) charges in the CFT are thus no longer R-charges, and therefore there is no BPS condition involving them. Spectral ow remains a robust property of the CFT that follows from symmetry, and leads to the same result as the combination of the decoupling limit of the BPS formula for the right-movers and the moduli space considerations employed in [17] to obtain the charge spectrum. This gives us further con dence in the applicability of these formulae, though with the caveat that the full energy of any given state will typically not be saturated by the contributions of the U(1) charges. A T4 symmetric product accounts for the rst term in (7.7) via the left-moving T 4 translation currents, and similarly the second and third terms correlate with the corresponding terms in (7.5). This leaves two additional left-moving currents of level N . The three \special" currents not associated to torus translations (two left-moving and one rightmoving), plus the right-moving R-symmetry current, thus all have level N and soak up all the central charge of the right-moving fermions in the symmetric product, and the corresponding remaining central charge of the left-movers. Bosonizing all four currents leads to a (2,2) lattice of zero modes whose energetics must match (7.5), (7.7). The energies of a general (2,2) lattice of zero-modes has the form h L# = R# = 1 1 18In comparing equation (7.5) above to the spectrum equation 5.24 of [63], one notes a typo of a missing factor of 1/2 in the rst term on the r.h.s. of the latter. for complex = 1 + i 2, = 1 + i 2. Without loss of generality, we can write 1 2 m2 = (mL + mR) ; n2 = (mL mR) 1 2 (7.9) (7.10) (7.11) (7.12) (7.13) HJEP06(217)3 and interpret mR as the eigenvalue of J R3 of the R-symmetry. Demanding that mR appear only in hR and only quadratically implies reproduced for 1 = 1 = 1, 2 = 2 = p 3 . The right-moving energetics (7.5) is R# = m2R + 4 (mL + 4m1 n1)2 12 mL + 4m1 n1 = d1 + p + d5 6789N : Examining the contribution of the charges to the left- and right-moving energies, the closest match comes if we identify mL = p5 f5 ; mL n1 = p5 + f5 ; 4m1 = d5 6789N which leads to a match between the lattice and supergravity expressions for the rightmoving energy. The di erence between the supergravity and symmetric product formulae then becomes hL L# = (N d5 6789)2 4 (N d5 6789)p5 2 which is reminiscent of the structure of a spectral ow. The low-lying spectrum (energies much less than order N ) is only compatible with d5 6789 = 0. The lattice of such states, when chosen to match the results of the BPS mass formula, cannot simultaneously accommodate the spectrum of free fermion superpartners of the torus translation currents. To summarize, supersymmetry and a symmetric product of c = 6 building blocks leads to a lattice of U(1) charges which is not compatible with the lattice inferred from supergravity considerations. The right-moving fermions which are the superpartners of right-moving translation currents have R-charge 1/2 and dimension 1/2; on the other hand, that lattice of charges inferred from supergravity does not have such a state in its spectrum. This throws considerable doubt on the existence of a symmetric product orbifold locus in the moduli space of the MSW CFT. 8 Discussion Understanding the dynamics of multiple M5 branes has been one of the most challenging and interesting issues in string theory for quite a number of years. There has been a huge e ort in understanding how M5-brane theories can describe strongly-coupled gauge theories in four dimensions. Our purpose in this paper has been to study what should, perhaps, be one of the simplest avatars of the M5-brane eld theory: the (1+1)-dimensional MSW CFT that comes from wrappings of an M5 brane on a very ample divisor of a Calabi-Yau manifold. This seemingly simple CFT remains enigmatic, almost twenty years after it was rst shown to be able to encode microstate structure of four-dimensional black holes [15]. In this paper we have considered M5 branes wrapping 4-cycles in T6 and T 2 K3, but our resulting M-theory solutions can be trivially extended to compacti cations with a more general eld content. As we have discussed, part of the di culty in analyzing this CFT is that it does not seem to have any point in its moduli space with a canonical description in terms of betterunderstood conformal eld theories, such as a symmetric orbifold theory. However, one can use holographic methods to study this theory at strong coupling, and in this paper we have made signi cant progress in that direction: we have obtained explicit families of smooth, horizonless solutions of ve-dimensional supergravity that are dual to families of BPS states of the MSW CFT. We constructed these families of solutions to M-theory by deriving a map between them and a class of states of the D1-D5 CFT, described as smooth, horizonless solutions to six-dimensional supergravity. This was done by transforming asymptotically AdS3 T4, D1-D5-P superstratum solutions that are independent of the Hopf ber of the S3 to asymptotically AdS3 S 2 T6 solutions19 dual to momentum-carrying microstates of the MSW CFT. We therefore referred to our new families of solutions as M-theory superstrata. S 3 In principle, one should be able to obtain families of M-theory superstrata that depend on arbitrary functions of two variables (with arbitrary Fourier modes around the axis of the S2 and the spatial axis of the AdS3). In this paper we have constructed solutions which have single Fourier mode excitations. However, based upon the success of the superstratum program in six dimensions [11], we anticipate that one should be able to nd smooth, horizonless M-theory superstrata with general families of Fourier modes excited. It is important to emphasize that there are many more M-theory superstrata solutions constructed using our technology than those that we have directly mapped to smooth D1D5-P superstrata. As we have seen in section 5, when the KKM charge, , is greater than one, the smooth D1-D5-P superstrata map to M-theory superstrata with mode numbers along the AdS3 circle that are multiples of . However, once in the M-theory frame, nothing prevents us from extrapolating these solutions to generic values of the mode numbers compatible with smoothness and appropriate M-theory periodicities. Under our map these more generic M-theory superstrata do not transform into geometric D1-D5-P states,20 and yet they are perfectly good solutions.21 As we have noted in the Introduction, there have been several earlier approaches to the construction of solutions dual to momentum-carrying BPS microstates of the MSW CFT. The common goal of this paper and of previous work has been to examine the spacetime structure of the microstates of black holes with a macroscopically-large horizon area. In 19Our solutions can trivially be extended by replacing T4 by K3 and T6 by T2 K3. 20A naive application of our map would give rise to solutions with multivalued elds, and if one extrapolates the candidate dual CFT states of [13, 14] to the appropriate values of the parameters, one would not satisfy the condition of integer momentum per strand. Thus a straightforward application of this holographic dictionary suggests that these con gurations should be discarded. For more discussion, see [13]. U-dualizing. 21Rather than using our map, one could also obtain these solutions by setting = 1 in the D1-D5 superstrata, restricting to k = 2m, introducing the smooth Z quotient of the Hopf ber by hand and this system, these black holes have a momentum charge along the AdS3 circle that, for given M5 charges, must be larger than a certain threshold, which is of order the product of the three M5 charges; once above this threshold, one is in the \black hole regime" of parameters. In Type IIA, the M5 and momentum charges become D4 and D0 charges. The microstate geometry corresponding to the maximally-spinning Ramond ground state of the MSW CFT is obtained by blowing up the single-center D4-D4-D4 con guration to a two-center uxed D6-D6 con guration, whose M-theory uplift is [global AdS3] S The addition of D0 charge via back-reacted singular D0's was studied in [22, 47]. The 2 T6. degeneracy of such \D0-halo" solutions was counted in [24, 25], and found to give rise to an entropy that matches that of an M-theory supergraviton gas in [global AdS3] for su ciently small D0 charge. The full back-reaction of the supergraviton gas states has never been computed, but since our M-theory superstratum solutions represent smooth waves in AdS3 S2, one may expect that at least some of them can be thought of as coming from back-reacted supergraviton gas states. Furthermore, if the full non-Abelian and nonperturbative interactions of uplifted D0-branes results in solutions that are nonsingular and varying along the M-theory circle, one expects these solutions to also resemble our M-theory superstrata. Hence, it may be that the smooth back-reacted solutions we construct are the missing link needed to connect the entropy counts in the (non-backS 2 T HJEP06(217)3 reacted) supergraviton gas and (singular) D0-halo approaches. In the black-hole regime of parameters, the D0-halo entropy exhibits a sub-leading growth with the charges compared to the black hole entropy [47]. On the other hand, in this regime the solutions have large deep AdS2 throats with high redshifts, and so are no longer small perturbations of [global AdS3] S 2 T6. A robust estimate of the number of states comprised by superstrata remains to be carried out. One can also add momentum by adding M2-branes that wrap the two-sphere of the AdS3 S2, and that carry angular momentum on both the AdS3 and the S2 [22]. The entropy of these con gurations comes from the high degeneracy of the Landau levels that result from the dynamics of the M2-branes on the compacti cation manifold in the presence of M5 ux [22, 66, 67], and has been argued to scale in the same way as that of the black hole. The back-reaction of these \W-brane" con gurations is fully worked out only in some very simple examples [30]. However, on general grounds one expects uncancelled tadpoles which give rise to asymptotics that are di erent from the asymptotics of bulk duals of MSW CFT states. If, on the other hand, one cancels the tadpoles using additional brane sources, there are no preserved supersymmetries whatsoever [31]. Furthermore, in more generic multi-center solutions, the corresponding W-brane con gurations also give rise to tadpoles, which can only cancel when the W-branes form a closed path among the centers.22 Hence, when the multi-center solution has a throat of nite length, these additional M2-brane bound states break at least another half of supersymmetry (giving 116 -BPS states), and typically all of the supersymmetry [31]. Thus these states cannot correspond to microstates of the BPS MSW black hole. 22The counting of these closed paths gives an entropy that scales in the same way as the black hole entropy as a function of the charges [67{69]. Given the large entropy of the W-branes, one would like to somehow restore the broken supersymmetry. This can only be achieved by going to a scaling limit, in which the throat becomes in nitely deep. In the in nite-throat limit, the solitonic W-branes become massless, new dynamical elds emerge (corresponding to the Higgs branch of the eld theory for which the W-branes are individual quanta) and the rich families of W-branes become re ections of the rich degeneracies of the vacua of these new dynamical elds. Thus W-branes should provide a semi-classical way to access the Higgs branch [67]. Another way to access the physics of the Higgs branch is via world-sheet disk amplitudes. Using these techniques one can compute the supergravity back-reaction of D-brane bound states upon an in nitesimal displacement on the Higgs branch. In the D1-D5 system, such calculations demonstrate that the additional tensor multiplet described by (Z4; G4) is an integral part of the back-reaction of generic Higgs-branch states [33]. Thus one expects the con gurations that result from condensing the W-branes to include such additional species of supergravity elds. For four-charge black holes in four dimensions, in the D3D3-D3-D3 system (which is U-dual to the D1-D5-KKM-P and the M5-M5-M5-P systems), a similar string emission calculation was recently performed [34, 70], con rming the presence of this kind of additional species of supergravity elds in the backreaction of these bound states. Remarkably, these new species of supergravity elds are exactly those needed to give rise to smooth superstrata solutions, via the coi uring procedure we have used in sections 5 and 6. Furthermore, if one back-reacts M5 branes of [15] wrapping smooth ample divisors inside T6, one expects to source exactly these additional supergravity elds. If one combines these two features with the fact that our M-theory superstrata solutions should be parameterized by arbitrary continuous functions, and hence have a large entropy, it appears very likely that these supergravity elds are a key component of the structure of typical black hole microstates. Our results raise some interesting questions about the formal mathematical structures of ve-dimensional supergravity solutions. One should recall that the construction of smooth microstate geometries in ve dimensions was done via locally hyper-Kahler base metrics whose signature changes from +4 to 4 on certain hypersurfaces. These singular base metrics are referred to as ambi-polar or pseudo-hyper-Kahler base metrics, and the hypersurfaces where the signature changes are referred to as \degeneration loci". While the four-dimensional spatial base metric is singular, all singularities cancel in the vedimensional Lorentzian metric. There has been a growing mathematical interest in the geometry of these ambi-polar spaces [71], generalizing the notion of \folded" hyper-Kahler metrics [72, 73]. Our results here indicate that harmonic analysis on such manifolds might be extremely rich and interesting. In particular, the rst step in solving the BPS equations is to nd smooth, harmonic two-forms on the spatial base metric. In standard Riemannian geometry, this is a classical exercise and the harmonic forms are dual to the homology cycles. The original work on microstate geometries involving ambi-polar bases [20, 21, 35] simply translated the expressions for the standard harmonic forms of Riemannian geometry. The solutions constructed in this paper have only one homology cycle, but we have exhibited in nitely many \pseudoharmonic" two-forms. We de ned such two-forms to be those that are closed and co-closed (\harmonic"), potentially singular on the degeneration loci of the base geometry, and yet lead to completely regular, ve-dimensional BPS solutions. This leads to several interesting questions. Firstly, how does our result generalize to multi-centered ambi-polar GH metrics? More generally, what is the classi cation of pseudo-harmonic two-forms? This paper shows that what seems to be a rigid topological problem actually has an in nite amount of \wiggle room" on an ambi-polar base. Returning to our map between states of the MSW and D1-D5 CFT's, the results presented here suggest that this map should contribute more deeply to our understanding of the physics of four-charge black holes in four dimensions and to the question of how much entropy of these black holes comes from smooth horizonless solutions. More broadly, we believe that our map will also prove useful in gaining deeper understanding of the hitherto mysterious MSW CFT. As we have seen, only a particular class of the MSW microstate geometries are related to D1-D5-P ones, and hence only a sub-sector of the states of the MSW CFT is mapped to a sub-sector of the D1-D5 CFT. It would be extremely interesting to explore and test possible extensions of this correspondence. Indeed, several important questions remain about our map. First, the map is de ned in terms of geometrical data, and it is interesting to see whether one can generalize it to other CFT states that are not dual to smooth torus-independent horizonless supergravity solutions, but may involve string or brane degrees of freedom, dependence on the internal directions, or high-curvature corrections.23 A pessimistic possibility is that our construction is merely an approximation that relates particular geometrical solutions in the supergravity limit, but that does not map CFT physics beyond small perturbations around the particular states that can be related to each other. On the other hand, it is tempting to speculate that, if one accepts holography as a correct description of all physics in asymptoticallyAdS backgrounds (including all 1=N corrections), a generalization of our map to degrees of freedom beyond six-dimensional supergravity may exist, and it would be interesting to investigate its properties. A related question is whether our map is simply a useful device for counting and classifying certain MSW states, or whether it is capable of capturing other CFT data such as anomalous dimensions or three-point functions. In the supergravity approximation, these quantities can in principle be computed perturbatively around a given solution [74], giving one hope that additional information about the MSW CFT could be gleaned. One can reasonably expect at least some three-point functions to be mapped from one sector of one CFT to another sector of the other CFT, because of non-renormalization theorems [75]. However, if one considers the four-point functions of an MSW operator that gets mapped to a D1-D5 one, these four-point functions are computed by summing over all operators in the intermediate channel, which may not belong to the relevant sub-sectors. 23For example, one can imagine constructing ten-dimensional supergravity solutions dual to D1-D5 microstates that have a non-trivial dependence on the torus coordinates, and therefore cannot be described in a six-dimensional truncation. Our map would take these solutions into holographic duals of MSW microstates that contain an in nite tower of Kaluza-Klein modes, and thus cannot be described in ve-dimensional supergravity. Furthermore, generic four-point functions are not protected when one deforms away from the free orbifold point of the D1-D5 CFT to the supergravity point, and hence there is no reason to expect a map for this data. Nevertheless, one might hope to use our map to nd a prescription that allows one to calculate at least certain conformal blocks of the MSW CFT from D1-D5 ones, which would already be remarkable progress. What is clear is that, as a CFT-to-CFT map, our construction is quite unusual. Indeed, to go from the D1-D5 NS vacuum to the MSW one, one needs to perform a combination of spectral ow transformations and \gauge" transformations. While spectral ow transformations have a clear CFT interpretation, as the rede ning of the CFT Hamiltonian by the addition of a term proportional to the R-charge, the gauge transformation would appear to correspond to rede ning the R-charge by the addition of a term proportional to the Hamiltonian, which is much more mysterious. Hence, while spectral ow is an operation that maps states to states within the CFT, the gauge transformation appears to change the CFT itself. On the other hand, since the MSW CFT does not appear to have any point in its moduli space with a symmetric product orbifold description, a map of the type we have found may be the most one can hope for. Acknowledgments We thank Massimo Bianchi, Duiliu Emanuel Diaconescu, Stefano Giusto, Monica Guica, Stefanos Katmadas, David Kutasov, Jose Francisco Morales, Rodolfo Russo, Masaki Shigemori and Amitabh Virmani for valuable discussions. EM and NPW are very grateful to the IPhT, CEA-Saclay for hospitality during the initial stages of this project. The work of IB and DT was supported by the John Templeton Foundation Grant 48222 and by the ANR grant Black-dS-String. The work of NPW was supported in part by the DOE grant DE-SC0011687; that of EJM was supported in part by DOE grant DE-SC0009924. The work of DT was further supported by a CEA Enhanced Eurotalents Fellowship. Covariant form of BPS ansatz and equations To rewrite our ansatz in covariant form, we rescale (Z4; 4; G4) ! (Z4; 4; G4)=p2. Then we have C123 = 1 ; C344 = 1 : It should be understood that this rescaling holds throughout this appendix (and only in this appendix). Then we de ne the (mostly-minus, light-cone) SO(1; 2) Minkowski metric via ab = C3ab 12 = 21 = 1 ; 33 = 1 : which can be used to raise and lower a; b indices, now that the above rescaling has been done. After the rescaling we have (A.1) (A.2) (A.3) P = 2 1 abZaZb = Z1Z2 12 Z42 : The rst layer of the BPS equations then takes the form 4DZ_a = abD (b) ; D 4 DZa = ab (b) ^ d ; (a) = 4 (a) : (A.4) The second layer becomes = Za (a) ; 4 ab 4 ^ (b) : Our ansatz for the tensor elds is G(a) = d P 2 1 abZb (du + !) ^ (dv + ) (dv + ) ^ In our conventions, the twisted self-duality condition for the eld strengths is 6 G(a) = M abG(b) ; Mab = ab : ZaZb P B Circular D1-D5 supertube: parameterizations In this appendix we recall the usual representation of a circular D1-D5 supertube solution within the six-dimensional ansatz (2.2), and the relation to the representation used in this paper. The usual representation (see, for example, [11, 13]) is given using the coordinate transformation (2.3) and setting F = 0. This solution is then given by: Z2 = Q5 Z4 = 0 ; (j) = 0 ; j = 1; 2; 4 ; Rya2 p 2 !e = ! + p 2 (A.5) (A.7) v~ = t + y ; Z2 = Q5 Fe = 1 ; (j) = 0 ; j = 1; 2; 4 ; Rya2 sin2 d'1 : (B.2) Z1 = u~ = t ; Z1 = Rya2 This is the form of the solution used in the main text in (3.4). asymptote to the form (B.2). For superstratum solutions, the same rede nitions can be applied, and then the elds C Lattice of identi cations and fractional spectral ow In this appendix we illustrate the step of rede ning the lattice of identi cations, with the explicit example of fractional spectral ow of a multi-wound circular D1-D5 supertube solution [10]. (sin2 d'1 (sin2 d'1 + cos2 d'2) : (B.1) Using the transformations (2.15) and (2.16), we obtain the following solution: Equivalently, one can use the coordinate form of this fractional spectral ow transformation, v^ v^ + 2 Ry ; ^ + 4 : = ^ s R v^ ; v = v^ : and act with it on the explicit multi-wound circular supertube metric (3.12), again impos Since ^ is the only coordinate that has transformed non-trivially, for ease of notation we shall re-use the coordinates of the starting solution , , t~, y~ de ned in (3.11), as well as , without writing hats explicitly. Then the transformed decoupling-limit solution is: h cosh2 dt~2 + d 2 + sinh2 dy~2i 4 d ^ (2s + 1)(dt~+ dy~) + cos d + (dt~ 2 d + (dt~ dy~) 2 : (C.1) (C.2) (C.3) (C.4) The starting con guration is the multi-wound circular D1-D5 supertube in the decoupling limit, given in eqs. (3.1){(3.5). To this solution we apply a (fractional) spectral ow transformation (2.29) with parameters ( 1; 2 ; 3) = (0; 0; s=( R)), together with an accompanying gauge transformation. The details and the resulting harmonic functions can be found in appendix A of [10]. Recall that we have de ned R Ry=2. The point that we emphasize here is that to generate the transformed solution, one inserts these new harmonic functions into a \hatted" version of the general ansatz, as in (2.21), and importantly, one takes the lattice of identi cations to be the standard one in the hatted coordinates: namely ing (C.1). As in the starting solution (s = 0), there is a coordinate change to bring the metric to local AdS3 S3 form. For the above transformed solution, it is of course 0 = ^ (2s + 1)(t~+ y~) ; 0 = + (t~ y~) : The combination of (C.1) and (C.4) gives rise to an interesting variety of orbifold singularities in the core of these solutions, depending on the common divisors of the integer parameters s, s + 1 and , as noted in [76] and analyzed in detail in [10]. D D.1 MSW maximal-charge Ramond ground state solution Coordinate conventions We record here for convenience some of our coordinate conventions. We de ne the threedimensional distances r from the centers, in Cartesian and cylindrical coordinates: qy12 + y22 + (y3 p 2 + (z c)2 ; c = 1 2 a : 8 (D.1) We de ne angular coordinates measured from the z = y3 axis at the two centers via cos # z : The relation between these coordinates and the (r; ) coordinates used throughout the paper is 4r 4r+ (r2 + a2 sin2 ) ; 1 2 1 cos # cos 2 #+ = 1=2 cos ; 1=2 sin ; 1=2 sin : Prolate spheroidal coordinates centered on r = 0 are useful for writing the metric as z = c cosh 2 cos ; = c sinh 2 sin ; 0 ; 0 : = c (cosh 2 cos ) : D.2 MSW maximal-charge Ramond ground state solution The ve-dimensional MSW maximal-charge Ramond ground state solution is described by the following harmonic functions. Using I = 1; 2; 3 and employing notation mod 3 for the I indices, we have V = LI = 1 1 4 kI+1kI+2 1 1 KI = M = 1 k k k 1 2 3 8 1 1 ; 1 k k k 1 2 3 2c : The four-dimensional base metric can be written as We write the one-form A as ds42 = V 1 d + A)2 + V (d 2 + dz2 + 2 d 2) : A = (cos #+ cos # )d : Note that in this gauge, near the GH centers, A ' ( 1 identi cations that gives smoothness is (cf. Footnote 12) cos # )d , so the lattice of The one-form $ is given by = + 4 ; + 2 : $ = k k k 1 2 3 4c 2 + (z c + r+)(z + c r ) r+r d : (D.2) (D.4) (D.6) (D.8) (D.9) (D.10) (D.11) The metric of this solution is that of global AdS3 S2. 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Iosif Bena, Emil Martinec, David Turton, Nicholas P. Warner. M-theory superstrata and the MSW string, Journal of High Energy Physics, 2017, 137, DOI: 10.1007/JHEP06(2017)137