Microstate Geometries from Gauged Supergravity in Three Dimensions
aa r X i v : . [ h e p - t h ] J un IPHT-T20/037
Microstate Geometries from Gauged Supergravityin Three Dimensions
Daniel R. Mayerson , Robert A. Walker , and Nicholas P. Warner , , Universit´e Paris Saclay, CNRS, CEA,Institut de Physique Th´eorique,91191, Gif sur Yvette, France Department of Physics and Astronomyand Department of Mathematics,University of Southern California,Los Angeles, CA 90089, USA daniel.mayerson @ ipht.fr, walkerra @ usc.edu, warner @ usc.edu
Abstract
The most detailed constructions of microstate geometries, and particularly of superstrata,are done using N = (1 ,
0) supergravity coupled to two anti-self-dual tensor multiplets in sixdimensions. We show that an important sub-sector of this theory has a consistent truncationto a particular gauged supergravity in three dimensions. Our consistent truncation is closelyrelated to those recently laid out by Samtleben and Sarıo˘glu [1], which enables us to developcomplete uplift formulae from the three-dimensional theory to six dimensions. We also find anew family of multi-mode superstrata, indexed by two arbitrary holomorphic functions of onecomplex variable, that live within our consistent truncation and use this family to provideextensive tests of our consistent truncation. We discuss some of the future applicationsof having an intrinsically three-dimensional formulation of a significant class of microstategeometries. ontents U (1) truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 , m, n ) superstrata . . . . . . . . . . . . . 224.3 The solutions in U (1) truncations . . . . . . . . . . . . . . . . . . . . . . . . . . 24 C.1 Consistency of gauge field actions, Chern-Simons term and conventions . . . . . 28C.2 Matching the uplift formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 (1 , m, n ) Superstrata in six dimensions 32
D.1 Six-dimensional BPS equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33D.2 The solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34D.3 Tuning the asymptotic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 36D.4 Regularity and CTC analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37D.5 Conserved charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38D.6 Uplifting the six-dimensional scalars to ten dimensions . . . . . . . . . . . . . . . 39
E Three-dimensional equations of motion 40
The construction of BPS/supersymmetric microstate geometries in five and six dimensionsis now a well-developed art [2–8]. In particular, superstrata represent one of the broadestfamilies of such geometries and have the advantage of a highly developed holographic dictio-nary [9–14, 4–6, 15]. Superstrata are based on the D1-D5 system, whose underlying CFT is cre-ated by open strings stretched between the branes, and so the field theory has a world-volumealong the common directions of the branes. The most general known families of superstrataare supersymmetric and encode a variety of left-moving excitations of the CFT. Encoding suchmomentum waves in the dual geometries means that they necessarily depend non-trivially onfive of the six dimensions. The construction of these geometries is only possible because of thedramatic simplification afforded by the linear structure of the BPS equations and the decom-position of the solution into its “linear pieces” [16]. Once these pieces are reassembled into thecomplete geometry, the metric appears to be remarkably complex, as it must be to encode allthe physical data of the underlying CFT states.One of the remarkable features that has become evident in recent constructions ofasymptotically-AdS superstrata [5, 17, 7] is that most of the interesting physics of superstratais encoded in a three-dimensional space-time, K . Indeed, the six-dimensional space-time of asuperstratum naturally decomposes into the S surface around the branes, the radial coordi-nate, r , and the common directions, ( t, y ) along the branes. The manifold, K , is the geometrydescribed by the coordinates ( t, y, r ) that are complementary to the S .Since we are working with the holographic dual of a (1 + 1)-dimensional CFT, the geometryis asymptotic to AdS × S , and the vacuum is simply global AdS × S . Superstrata involveturning on new fluxes and adding metric deformations, thereby creating a warped, fibered prod-uct, K × S . The manifold, K , is a smooth, horizonless, three-dimensional space-time and the S is usually deformed and fibered over K by non-trivial Kaluza-Klein Maxwell fields.The manifold, K , is best described as a “smoothly-capped BTZ geometry.” That is, like The remaining directions of the D5 are compactified on the T that reduces IIB supergravity to six dimensions. at infinity and has a long AdS × S throat, but unlike BTZ,this throat has a finite depth because it caps off smoothly without a horizon. These geometriesthus look much like the horizon region of a black hole, except that there there is a finite redshiftbetween the cap and any point in the asymptotic region. It is these three-dimensional geometriesthat have provided the basis of many of the recent studies and comparison between microstategeometries and black holes [18–22].The analysis of such microstate geometries was greatly facilitated by the fact that, for somesuperstrata, the massless scalar wave equation in six dimensions is separable [17,23,22], reducingto a simple Laplacian on a “round” S and a far more complicated wave equation on K . Forthese geometries, the physics of massless scalar waves could indeed be entirely reduced to aproblem on K . It was also conjectured, based on indirect evidence, in [17] that some superstratashould be part of a consistent truncation to a gauged supergravity in three dimensions.The purpose of this paper is to prove this conjecture by showing that the six-dimensionalgauged supergravity that is the “work-horse” of superstrata construction, does indeed have aconsistent truncation down to a three-dimensional gauged supergravity. We will also give someexplicit superstrata solutions that are entirely captured by this truncation.Consistent truncations have a long history in supergravity and we will not review this here.There are the relatively trivial consistent truncations that are based on reducing a higher-dimensional supergravity on a manifold that has isometries and restricting fields to singletsof those isometries. This includes all the standard torus compactifications. There are alsohighly non-trivial consistent truncations that involve sphere compactifications in which one keepshigher-dimensional fields that depend (at linear order) on particular sets of “lowest harmonics”on the sphere. These fields therefore, typically, transform non-trivially under the rotation groupof the sphere. The isometries of the sphere also give rise to a non-abelian gauge symmetry inthe lower dimension. The end result is a compactification that reduces a sector of the higher-dimensional supergravity to gauged supergravity in lower dimensions. Here we will be concernedwith S compactifications of six-dimensional supergravity coupled to some tensor multiplets, andthe corresponding three-dimensional gauged supergravity theory. We will also show the consis-tent truncation encodes some rich families of superstrata, some of which have been constructedelsewhere [17, 7].We also construct new families of superstrata that depend on two freely-choosable holomor-phic functions and that live entirely within our consistent truncation.An important point about consistent truncations is that they are not merely lower-dimensional effective field theories. If one solves the lower-dimensional equations of motionin a consistent truncation, the result is an exact solution of the higher-dimensional equations ofmotion. This fact can be immensely useful in simplifying the equations of motion. In particular,the sphere becomes an “auxiliary” space whose dynamics is entirely determined by the lower-dimensional theory and encoded in the details of the consistent truncation. In this way, one canreduce a higher dimensional problem to a much more tractable lower-dimensional problem.Consistent truncations can prove to be a ‘Faustian Bargain.’ The price of the simplificationis a huge restriction on the degrees of freedom: the higher dimensional theory has vastly more4egrees of freedom than the lower-dimensional theory and these extra degrees of freedom mayprove essential to capturing the correct physics. The study of holographic (2 + 1)- and (3 + 1)-dimensional field theories is littered with examples in which consistent truncations have capturedthe essential physics, as well as examples in which the consistent truncation has lacked thenecessary resolution to produce the correct physics. We will discuss this further in Section 5.We have several reason for constructing the consistent truncations that are relevant to su-perstrata.First, motivated by the success of such a strategy for holographic field theories in (2 +1) and (3 + 1) dimensions, we wish to mine everything that three-dimensional supergravitieshave to tell us about holographic field theories in (1 + 1)-dimensions, and the correspondingsupergravity solutions in in six dimensions. Again, the lower dimensional BPS equations aremuch simpler than the higher-dimensional BPS equations, since solutions are functions of 2rather than 5 variables, and thus may yield extremely interesting new holographic flows. Thethree-dimensional formulation may also lead to a deeper understanding of the moduli spaceof superstrata and the microstates they represent. For example, we know, from perturbationtheory [24, 25], that there are supersymmetric metric perturbations of superstrata. As yet, wedo not know how to “integrate” these perturbations up to finite moduli and thereby create newfamilies of superstrata. It is possible that the three-dimensional formulation will simplify a classof these moduli and show us how to do this more generally.Above all, is the possibility of getting a handle on non-supersymmetric, non-BPS superstrata.Given the intrinsic complexity of even the supersymmetric superstrata in six-dimensions, itseems an overwhelming task to address the non-linear equations that necessarily underlie theconstruction of non-BPS superstrata. Indeed, such generic non-BPS superstrata are expectedto depend non-trivially on all six dimensions. However, the consistent truncation we present inthis paper reduces this problem, for some limited families of superstrata, to a three-dimensionalproblem. Solving the equations of motion for the three-dimensional supergravity will still be aformidable task, and we intend to explore this in future work. The importance of the resultspresented here is that they transform an impossible six-dimensional problem into a feasiblethree-dimensional problem. Paper overview
In Section 2, we describe the class of three-dimensional gauged supergravity theories that can en-code superstrata; a summary of the supergravity theory, fields, and action is given in Section 2.7.The details of how this theory uplifts to six-dimensional supergravity may be found in Section3. Specifically, we show how the consistent truncation works: how the three-dimensional fieldsare encoded in the six-dimensional supergravity and how the solutions of the three-dimensionalequations yield a solution to the six-dimensional equations. In Section 4 we describe a newclass of six-dimensional BPS superstrata (whose computational details may be found in Ap-pendix D) that fit within the consistent truncation described in Section 2. We reduce thesesix-dimensional solutions to their three-dimensional data and use them to test the details of theconsistent truncation. The BPS superstrata that we have construct are intrinsically new in that5uch a multi-function family, while in similar spirit to those in [7], have not been constructedbefore.In Section 5 we make some final remarks and return to the discussion of the applications ofour results.
In this section, we will discuss a specific three-dimensional gauged supergravity theory whichis relevant for the dimensional reduction of six-dimensional superstrata. The summary of ourresulting three-dimensional theory is given in Section 2.7.
If one reduces IIB supergravity on T , one obtains the N = (2 ,
2) theory in six dimensions.Reducing on a K
3, instead, halves the supersymmetry to those that are holonomy invariant, andthe result is an N = (2 ,
0) supergravity theory coupled to 21 anti-self-dual tensor multiplets.More generally, the “parent theories” of interest here are six-dimensional N = (2 ,
0) su-pergravity (with sixteen supersymmetries) coupled to n tensor multiplets. In such theories,the graviton multiplet contains one graviton, two complex, left-handed gravitinos (or foursymplectic-Majorana Weyl gravitinos) and five self-dual, rank-two tensors gauge fields. Eachtensor multiplet contains one anti-self-dual, rank-two tensor gauge fields, two right-handed com-plex spinors (or four symplectic-Majorana Weyl spinors) and five real scalars. The R -symmetryis SO (5) ∼ = U Sp (4), the tensor gauge fields transform in the fundamental of SO (5 , n ) and thescalars are described in terms of a coset: SO (5 , n ) SO (5) × SO ( n ) . (2.1)In the fully non-linear theory, the scalar matrix plays an essential role in a twisted dualitycondition on the tensor gauge fields. We will discuss a reduced version of this below.This six-dimensional supergravity can then be compactified on AdS × S using a ‘Freund-Rubin’ Ansatz in which one of the self-dual field strengths is set equal to the volume form ofAdS and of S . This corresponds to the D1-D5 background in which the supergravity charges, Q and Q , are set equal. If one wants unequal charges one must move some flux into ananti-self-dual tensor gauge field. The simple, self-dual flux breaks the SO (5 , n ) symmetry to SO (4 , n ).There is now an extensive literature [26–31, 1] on how this compactification leads to N = 8(16 supersymmetries) gauged supergravity in three dimensions. The gauge group is SO (4) ∼ =6 O (3) + × SO (3) − and comes from the isometries of S ; the scalar coset becomes : SO (8 , n ) SO (8) × SO (3 + n ) . (2.2)The gauge group sits inside SO (8 , n + 3) as the diagonal SO (4) in the first and third factors ofthe decomposition [30]: SO (4) × SO (4) × SO (4) × SO ( n − ⊂ SO (8 , n ) . (2.3)In particular, the precise relationship between N = (2 ,
0) supergravity coupled to one anti-self-dual tensor multiplet in six dimensions and the three-dimensional N = 8, SO (8 ,
4) supergravitywas recently laid out in [1].The construction of superstrata usually takes place in the less supersymmetric, N = (1 , T compactification of IIBsupergravity and then further restricting to only the fields that transform trivially under the SO (4) global rotations on the tangent space of the T . This results in N = (1 ,
0) supergravitycoupled to two anti-self dual tensor multiplets. To be precise, the ten-dimensional RR field, C (2) , descends to the self-dual tensor in the gravity multiplet and one of the two anti-self-dualtensors. These two components are independent and account for the separate D1 and D5 pieces.The other anti-self-dual tensor descends from the ten-dimensional Kalb-Ramond field, B (2) ,and anti-self-duality is required by supersymmetry. Roughly, the only way that this can becompatible with N = (1 ,
0) supersymmetry in a D1-D5 system is if the F1 and NS5 fields arelocked together via anti-self-duality.Imposing invariance under global rotations on T reduces the SO (5 , n ) coset in six-dimensionsto SO (1 , n ). The compactification to three dimensions on AdS × S then results in the scalarcoset SO (4 , n ) SO (4) × SO (3 + n ) . (2.4)The relationship between N = (1 ,
0) supergravity coupled to one anti-self-dual tensor multipletin six dimensions and the three-dimensional N = 4, SO (4 ,
4) supergravity (with n = 1) has beenlaid out in [26, 27, 32, 31].As we noted above, superstrata require N = (1 ,
0) supergravity coupled to at least twotensor multiplets. The extra tensor multiplet plays an essential role in the construction of Here we are going to consider the lowest KK towers in the compactification. Remarkably, it seems that onecan consistently truncate in a manner that allows higher modes in the KK towers [30]. As we will discuss inSection 5, this might prove immensely useful in using three-dimensional supergravity to construct much moregeneral classes of superstrata. mooth solutions. Indeed, string amplitude computations [33,34] and the holographic dictionaryshowed that it is inconsistent to freeze out this degree of freedom. It was this realization thatled to the first successful construction of non-trivial superstrata in [4].We also note that the three-dimensional supergravity corresponding to N = (1 ,
0) supergrav-ity coupled to one tensor multiplet was used in [35] to construct some black-ring and black-stringsolutions. The consistent truncation proved to be a useful tool, but the lack of the extra tensormultiplet meant that the solutions were singular and that superstrata were inaccessible fromwithin such a truncation.We are therefore going to examine the three-dimensional, (0 ,
2) supergravity (with eightsupersymmetries) for which the scalar coset is SO (4 , SO (4) × SO (5) . (2.5)As will become evident, we will find the results in [1] for N = (2 ,
0) supergravity immenselyuseful in extending the results of [31] to obtain the three-dimensional supergravity correspondingto N = (1 ,
0) supergravity coupled to two tensor multiplets.The relevant supergravity in three dimensions is fully defined by its amount of supersymme-try, the scalar coset, the gauge symmetry and the gauge couplings as defined by an embeddingtensor. The number of bosonic degrees of freedom in the theory is equal to the dimension ofthe underlying coset. However, these degrees of freedom can be encoded in various ways in theaction. In three dimensions, Yang-Mills gauge fields can be dualized into scalars and vice versaand this is how the Yang-Mills fields can be generated. In addition, one of the essential featuresof the three-dimensional theories is the appearance of massive Chern-Simons vector fields. Thesefields can be viewed as gauging non-semi-simple groups and can ultimately be integrated out.Thus, as explained in [29], the number of bosonic degrees of freedom, d , is given by: d = dim(Coset) = . (2.6)For the three-dimensional (0 ,
2) supergravity theories described above, with coset (2.4), thegauge group is actually a semi-direct product SO (4) ⋉ T ⊂ SO (4 , n ), where SO (4) is thestandard Yang-Mills gauge group coming from S and T is a translation that transforms in theadjoint of SO (4). Thus the 4 × (3 + n ) degrees of freedom become 6 YM vectors, 6 CS vectorsand 4 n scalars. To describe this theory we simply follow the discussion in [31] but with an extra tensor multiplet.The SO (4 ,
5) group has an invariant metric: η ≡ × × × ×
00 0 ε , (2.7)where ε = ±
1. Note that we are using the anti-diagonal form of η because it is far moreconvenient in describing the degrees of freedom and in expressing the gauging. For ε = − G = SO (5 ,
4) (in the conventions of [1]) and for ε = 1 we have G = SO (4 , ε = 1,whereas (a truncated version of) the theory in [1] corresponds to ε = − G may be written as: A B χ A C − A T λ A − ελ A − εχ A , (2.8)where A, B, C, D are 4 × B T = − B and C T = − C . The matrix A generates a GL (4 , R ) whose compact generators define the SO (4) YM gauge group and whose non-compactgenerators are obtained by taking A T = A . The remaining 10 non-compact generators can betaken to be B and χ A . That is, we will choose to parametrize the coset by setting A T = A , C = 0 and λ A = 0. The matrix B describes the translation generators of T transforming in theadjoint of SO (4). In this formulation, the 20 bosonic degrees of freedom are defined by A T = A , B T = − B and χ A .The simplest way to fix the T gauge invariance is to set B = 0, which we will now do. Wecould also fix the SO (4) gauge invariance by reducing A to a diagonal matrix. The 20 degreesof freedom would then be the 4 eigenvalues, the 4 χ A ’s and 6 + 6 gauge fields. However, we willonly go half-way: fixing the T gauge and moving these degrees of freedom into the CS vectors.We will preserve the SO (4) gauge invariance.Thus our scalar matrix will be defined by: V ¯ M ¯ K = exp χ A − εχ A P AB P − ) BA
10 0 0 = P AB − ε χ A (cid:0) ( P − ) BC χ C (cid:1) χ A P − ) BA − ε ( P − ) BC χ C , (2.9)where P = P T is a symmetric GL (4 , R ) matrix.Our index conventions will be as follows. A vector of G will be denoted by X ¯ M ≡ ( X A , X A , X ) , X ¯ M ≡ ( X A , X A , ε X ) , (2.10)where the indices are raised and lowered using (2.7). The components, X A and X A , transform,respectively, in the 4 and 4 of GL (4 , R ). That is, they transform through multiplication by P or P − , respectively.Following [1], we define: M ¯ A ¯ B = ( VV T ) ¯ A ¯ B = V ¯ A ¯ C V ¯ B ¯ C , M ¯ A ¯ B = (cid:0) ( V T ) − ( V − ) (cid:1) ¯ A ¯ B = (cid:0) V − (cid:1) ¯ C ¯ A (cid:0) V − (cid:1) ¯ C ¯ B ,m AB = ( P P T ) AB = P AC P BC , m AB = (cid:0) ( P T ) − P − (cid:1) AB = ( P − ) C A ( P − ) C B . (2.11)9 .3 The gauge couplings The embedding of the gauge group SO (4) ⋉ T in G is defined through the embedding tensor Θ.Specifically, if T ¯ M ¯ N = − T ¯ N ¯ M are the generators of G , then the covariant derivative is definedby: b D µ X ¯ P ≡ ∂ µ X ¯ P + A µ ¯ K ¯ L Θ ¯ K ¯ L, ¯ M ¯ N ( T ¯ M ¯ N ) ¯ P ¯ Q ( X ¯ Q ) , (2.12)where A µ ¯ K ¯ L are the gauge connections and ( T ¯ M ¯ N ) ¯ P ¯ Q ( X ¯ Q ) represents the action of T ¯ M ¯ N onvectors. In the standard normalization, one has:( T ¯ M ¯ N ) ¯ P ¯ Q ( X ¯ Q ) = δ ¯ N ¯ P X ¯ M − δ ¯ M ¯ P X ¯ N , (2.13)and these matrices have the commutation relations : (cid:2) T ¯ K ¯ L , T ¯ M ¯ N (cid:3) = f ¯ K ¯ L, ¯ M ¯ N ¯ P ¯ Q T ¯ P ¯ Q = η ¯ K ¯ M T ¯ L ¯ N + η ¯ K ¯ N T ¯ M ¯ L − η ¯ L ¯ M T ¯ K ¯ N − η ¯ L ¯ N T ¯ M ¯ K . (2.14)This defines the structure constants: f ¯ K ¯ L, ¯ M ¯ N ¯ P ¯ Q = η ¯ K ¯ M (cid:0) δ ¯ L ¯ P δ ¯ N ¯ Q − δ ¯ L ¯ Q δ ¯ N ¯ P (cid:1) + η ¯ K ¯ N (cid:0) δ ¯ M ¯ P δ ¯ L ¯ Q − δ ¯ M ¯ Q δ ¯ L ¯ P (cid:1) − η ¯ L ¯ M (cid:0) δ ¯ K ¯ P δ ¯ N ¯ Q − δ ¯ K ¯ Q δ ¯ N ¯ P (cid:1) − η ¯ L ¯ N (cid:0) δ ¯ M ¯ P δ ¯ K ¯ Q − δ ¯ M ¯ Q δ ¯ K ¯ P (cid:1) (2.15)in which indices are summed without any weight factors .The generic form of the embedding tensor is:Θ ¯ K ¯ L, ¯ M ¯ N = θ ¯ K ¯ L ¯ M ¯ N + (cid:0) η ¯ M [ ¯ K θ ¯ L ] ¯ N − η ¯ N [ ¯ K θ ¯ L ] ¯ M (cid:1) + θ η ¯ M [ ¯ K η ¯ L ] ¯ N , (2.16)where θ [ ¯ K ¯ L ¯ M ¯ N ] = θ ¯ K ¯ L ¯ M ¯ N and θ ¯ K ¯ L = θ ¯ L ¯ K . However, for the gauged SO (4) × T theory ofinterest here, the only non-vanishing pieces are [30, 31, 1]: θ ABCD = − α ǫ ABCD , θ
ABC D = γ ǫ ABCE δ DE , (2.17)for some coupling constants α and γ . It is in this expression that the GL (4 , R ) formulationarising from the choice (2.7) leads to significant simplification.In particular, this embedding tensor reduces A µ ¯ K ¯ L to the twelve independent gauge fieldsfor SO (4) ⋉ T : A µAB = − A µBA , A µBA = − A µ BA . (2.18)It is convenient to define: e A µAB ≡ ǫ ABCD A µCD , b A µAB ≡ ǫ ABCD A µC D , (2.19)and introduce: B µAB ≡ (cid:0) α e A µAB − γ b A µAB (cid:1) . (2.20) Our conventions, and the signs of the structure constants, differ from those of [1]. We discuss our choices andconsistent formulation of the gauge action in detail in Appendix C.1. This “double counts” the generators because T ¯ P ¯ Q = − T ¯ Q ¯ P . This is, however, a completely standard conven-tion that we use everywhere in this paper. GL (4 , R ) components as: b D µ X A = ∂ µ X A + B µAB X B − γ e A µAB X B , b D µ X A = ∂ µ X A − γ e A µAB X B , b D µ X = ∂ µ X . (2.21)Note that, in terms of the matrices of G , the connection B µAB has the form: B µ ≡ B µAB
00 0 00 0 0 . (2.22)These are therefore precisely the gauge fields of T . The vector fields A µAB are those of SO (4)but they act with their duals, and with a gauge coupling of − γ .To make this more explicit, define the SO (3) + × SO (3) − parts of the gauge connection, andits dual: A µAB = A + µ AB + A − µ AB , e A µAB = A + µ AB − A − µ AB , (2.23)and define the gauge couplings g + = − g − = − γ . (2.24)Then one has b D µ X A = ∂ µ X A + B µAB X B + g + A + µ AB X B + g − A − µ AB X B , b D µ X A = ∂ µ X A + g + A + µ AB X B + g − A − µ AB X B , b D µ X = ∂ µ X . (2.25)Finally, it is convenient to define the reduced, purely- SO (4), covariant derivatives: D µ X A = ∂ µ X A − γ e A µAB X B , D µ X A = ∂ µ X A − γ e A µAB X B , D µ X = ∂ µ X . (2.26) From [1], the scalar action is L scalar = 132 (cid:0) b D µ M ¯ K ¯ L (cid:1) (cid:0) b D µ M ¯ K ¯ L (cid:1) − V , (2.27)where the potential, V , is given by: V = 148 θ ¯ K ¯ L ¯ M ¯ N θ ¯ P ¯ Q ¯ R ¯ S (cid:16) M ¯ K ¯ P M ¯ L ¯ Q M ¯ M ¯ R M ¯ N ¯ S − M ¯ K ¯ P M ¯ L ¯ Q η ¯ M ¯ R η ¯ N ¯ S + 8 M ¯ K ¯ P η ¯ L ¯ Q η ¯ M ¯ R η ¯ N ¯ S − η ¯ K ¯ P η ¯ L ¯ Q η ¯ M ¯ R η ¯ N ¯ S (cid:17) + 132 θ ¯ K ¯ L θ ¯ P ¯ Q (cid:16) M ¯ K ¯ P M ¯ L ¯ Q − η ¯ K ¯ P η ¯ L ¯ Q − M ¯ K ¯ L M ¯ P ¯ Q (cid:17) + θ θ ¯ K ¯ L M ¯ K ¯ L − θ (2.28)Using the expressions above, we find the following result: L scalar = − Tr (cid:2)(cid:0) D µ m (cid:1) m − (cid:0) D µ m (cid:1) m − (cid:3) − m AB ( D µ χ A ) ( D µ χ B ) − m AC m BD (cid:16) B µAB − ε Y µAB (cid:17) (cid:16) B µCD − ε Y µ CD (cid:17) − V . (2.29)11here D µ is the SO (4) covariant derivative defined in (2.26), and where Y µ AB ≡ χ B D µ χ A − χ A D µ χ B , (2.30)and V = det (cid:0) m AB (cid:1) h (cid:0) α + ε γ ( χ A χ A ) (cid:1) + γ (cid:0) m AB (cid:0) m AB + χ A χ B (cid:1) − m AA m BB (cid:1)i . (2.31) The general Chern-Simons term is: L CS = 14 ε µνρ A µ ¯ K ¯ L Θ ¯ K ¯ L, ¯ M ¯ N (cid:16) ∂ ν A ρ ¯ M ¯ N + 13 f ¯ M ¯ N, ¯ P ¯ Q ¯ R ¯ S Θ ¯ P ¯ Q, ¯ U ¯ V A ν ¯ U ¯ V A ρ ¯ R ¯ S (cid:17) . (2.32)We have reversed the sign of the last term relative to [1] so as to have the canonical Chern-Simons action. (We discuss our choices and consistent formulation of the gauge action in detailin Appendix C.1.)Using the embedding tensor (2.17) and the structure constants (2.15), we find L CS = − ε µνρ h α (cid:0) A µAB ∂ ν e A ρBA − γ A µAB A νBC A ρCA (cid:1) − B µBA F ABνρ i , (2.33)where: F ABνρ ≡ (cid:0) ∂ [ ν A ρ ] AB − γ A [ νC [ A e A ρ ] B ] C (cid:1) . (2.34)When written in terms of SO (3) + and SO (3) − , we note that this action takes the more familiarChern-Simons form: L CS = − α ε µνρ h(cid:0) A + µ AB ∂ ν A + ρ BA + g + A + µ AB A + ν BC A + ρ CA (cid:1) − (cid:0) A − µ AB ∂ ν A − ρ BA + g − A − µ AB e A − ν BC e A − ρ CA (cid:1)i + ε µνρ B µBA F ABνρ , (2.35)with: F ABνρ = F + ABνρ + F − ABνρ , F ± ABνρ ≡ (cid:0) ∂ [ ν A ± ρ ] AB + g ± A ± [ νC [ A A ± ρ ] B ] C (cid:1) . (2.36) The Chern-Simons gauge fields, B µAB , appear only quadratically in the action, and withoutderivatives. It is therefore trivial to integrate them out by completing the square. The completebosonic action may be written L = R − Tr (cid:2)(cid:0) D µ m (cid:1) m − (cid:0) D µ m (cid:1) m − (cid:3) − m AB ( D µ χ A ) ( D µ χ B ) − V − m AC m BD F ABµν F µν CD − α ε µνρ (cid:0) A µAB ∂ ν e A ρBA − γ A µAB A νBC A ρCA (cid:1) − ε ε µνρ Y µAB F ABνρ + L B , (2.37)12here L B ≡ − g µν m AC m BD (cid:16) B µAB + ε µρ ρ m AE m BE F E E ρ ρ − ε Y µ AB (cid:17) × (cid:16) B νCD + ε νρ ρ m CE m DE F E E ρ ρ − ε Y νCD (cid:17) . (2.38)Thus the equations of motion for B µAB are trivial, and yield B µAB = ε Y µ AB − ε µνρ m AC m BD F CDνρ , (2.39)and these gauge fields drop out of the action entirely. The three-dimensional supergravity theory we consider is a (0 ,
2) gauged supergravity with eightsupersymmetries. It has an SO (4) gauge symmetry, with 6 gauge fields, A ABµ = A [ AB ] µ , where theindices A, B, . . . = 1 , , , SO (4). In addition to the graviton, andthe gauge fields, A ABµ , there are 14 scalar fields in the bosonic sector. Four scalars are encodedin an SO (4) vector, χ A , and the other ten are encoded as a general, symmetric GL (4 , R ) matrix, m AB = m ( AB ) , with inverse, m AB .The bosonic action is L = R − Tr (cid:2)(cid:0) D µ m (cid:1) m − (cid:0) D µ m (cid:1) m − (cid:3) − m AB ( D µ χ A ) ( D µ χ B ) − V − m AC m BD F ABµν F µν CD − α ε µνρ (cid:0) A µAB ∂ ν e A ρBA − γ A µAB A νBC A ρCA (cid:1) − ε ε µνρ Y µAB F ABνρ , (2.40)where m denotes m AB , and Y µ AB ≡ χ B D µ χ A − χ A D µ χ B . (2.41)The covariant derivative is defined on upper and lower SO (4) indices as D µ X A = ∂ µ X A − γ e A µAB X B , D µ X A = ∂ µ X A − γ e A µAB X B , (2.42)where e A µAB ≡ ǫ ABCD A µCD . (2.43)The field strengths are given by F ABνρ ≡ (cid:0) ∂ [ ν A ρ ] AB − γ A [ νC [ A e A ρ ] B ] C (cid:1) . (2.44)The scalar potential is: V = det (cid:0) m AB (cid:1) h (cid:0) α + ε γ ( χ A χ A ) (cid:1) + γ (cid:0) m AB (cid:0) m AB + χ A χ B (cid:1) − m AA m BB (cid:1)i . (2.45)The equations of motion following from (2.40) are given explicitly in Appendix E.The bosonic theory has three parameters: the gauge coupling, γ , a scale parameter, α , anda “signature,” ε = ±
1. In Appendix B, we discuss two scale invariances of the action that canbe used to set | α | = 1 and γ = 1. We will, however, retain these parameters.13he parameter ε , and the sign of α are extremely important to the supersymmetry. Wewill discuss this further in the six-dimensional context in Section 3.1. Here we will simply notethat sending ε → − ε, α → − α leaves the potential invariant. In the rest of the action, theparameters ε and α only appear as coefficients of the parity odd terms involving ε µνρ . Thus ε → − ε, α → − α , combined with an orientation reversal in the three-dimensional space-time isa symmetry of the action. Here, we will describe how the three-dimensional theory of Section 2 can be obtained from adimensional reduction from six dimensions. First, in Section 3.1, we discuss how we obtainedour consistent truncation formulae. In Section 3.2, we describe the six-dimensional theory athand, which is the theory relevant for the description of superstrata. In Section 3.3, we give thefull non-linear reduction ansatz from this six-dimensional theory to the three-dimensional theoryof Section 2. Finally, in Section 3.4, we describe a useful U (1) subsector of this truncation andits three-dimensional counterpart. The common core of superstrata and the consistent truncations of [26, 27, 32, 31, 1] is the basic N = (1 ,
0) supergravity coupled to single anti-self-dual tensor multiplet. This has an SO (1 , N = (2 ,
0) supergravity theory considered in [1] involves adding fourmore self-dual tensors to complete the (2 ,
0) graviton multiplet, as well as adding scalars toextend the anti-self-dual tensor multiplet to (2 ,
0) supersymmetry. This extends the SO (1 , SO (4 + 1 , ,
0) anti-self-dualtensor multiplets, which extend the SO (1 ,
1) to SO (1 , n ), where n is the number of addedanti-self-dual tensor multiplets.From the three-dimensional perspective, these two extensions of the six-dimensional theoryinvolves extending the SO (4 ,
4) of the consistent truncation of the basic theory [26, 27, 32, 31] to SO (4 + 4 ,
4) or to SO (4 , n ). The parameter, ε thus directly encodes whether we are addingself-dual or anti-self-dual tensors to the basic theory.As we noted earlier, flipping sign of ε and α along with a change of orientation, leaves thethree-dimensional bosonic equations of motion unchanged in three dimensions. The same isalso true for the six-dimensional equations of motion: such an orientation flip on the three-dimensional base only changes the duality conditions (3.1) in the six-dimensional bosonic equa-tions of motion. Thus, even in the six-dimensional theory, an SO ( p, q ) theory with p self-dualand q anti-self-dual multiplets, and the theory with p and q interchanged have exactly the samebosonic equations of motion (modulo orientations).In six-dimensional supergravity theories, there is a correlation between the chirality of thesupersymmetry and duality of the tensor gauge fields that belong to matter multiplets, or tothe graviton multiplet. The convention that is used for superstrata, and is used in [1], is thatthe self-dual tensors belong to the graviton multiplet and anti-self-dual tensors belong to matter14ultiplets. If one performed such an orientation flip on the (2 ,
0) theory, it would break thesupersymmetry (to (1 , SO (4 + 4 ,
4) theory, necessarily provide solutions to the equations of motion tothe six-dimensional theory with five self-dual tensors multiplets and one anti-self-dual tensormultiplet. A trivial orientation flip, means that this result maps onto the SO (4 , SO (4 , .Thus the existence of the consistent truncation we seek is already guaranteed by the resultsof [36, 1]. What remains is to adapt the uplift formulae of [1] to the theory of interest to us. Wewill also subject our truncation and uplift formulae to extensive and rigorous testing. The general N = (1 ,
0) six-dimensional supergravity theory coupled to an arbitrary number oftensor multiplets is discussed at length in [37, 38]. We will consider the N = (1 ,
0) supergravitymultiplet coupled to two anti-self-dual tensor multiplets, as this is the relevant sector thatcaptures the D1-D5-P solutions when reducing from ten-dimensional type IIB theory on a T [39](see also [40, 41]; or Appendix B of [42] for a quick summary). The six-dimensional bosonic field content we consider consists of the metric g ˆ µ ˆ ν , two scalars ϕ, X ,and three three-forms G ˆ I , ˆ I = 1 , , The theory has a SO (1 ,
2) global symmetry, where thetwo scalars parametrize a SO (1 , /SO (2) coset. The three-forms satisfy a self-duality relation:ˆ ∗ G I = 1(3!) ǫ ˆ α ˆ β ˆ γ ˆ µ ˆ ν ˆ ρ G ˆ I ˆ α ˆ β ˆ γ dx ˆ µ ∧ dx ˆ ν ∧ dx ˆ ρ = ε M ˆ I ˆ J G ˆ J , (3.1)where ε = ±
1, which serves as their equations of motion, together with their Bianchi identities: dG ˆ I = 0 . (3.2) This truncation is trivially achieved by by imposing an invariance under the SO (3) that acts on three of thefive anti-self-dual tensor multiplets. We conform to the idiosyncratic notation and conventions for the six-dimensional three-forms that is used inthe superstrata literature. This slightly odd notation of omitting the index 3 is historical. In reduction of thesix-dimensional system to five dimensions the F and dβ fields are identified with Z and Θ respectively.
15e have parametrized the scalar self-duality matrix M as: M ˆ I ˆ J = 12 e √ ϕ X − √ X e − √ ϕ (2 + e √ ϕ X ) X − X √ (2 e −√ ϕ + X )+ X √ (2 e −√ ϕ + X ) 4 √ X − e −√ ϕ − X , (3.3)Note that we use the following conventions for the SO (1 ,
2) metric which is used to raise orlower indices: η ˆ I ˆ J = − . (3.4)The other bosonic equations of motion can be obtained by varying the pseudo-Lagrangian [37,38]: L D = R −
12 ( ∂ ˆ µ ϕ ) − e √ ϕ ( ∂ ˆ µ X ) − M ˆ I ˆ J G ˆ I ˆ µ ˆ ν ˆ ρ G ˆ J ˆ µ ˆ ν ˆ ρ . (3.5)Note that the scalar matrix M ˆ I ˆ J (with both indices down) is symmetric.One should also note that the matrix M ˆ I ˆ J has one positive eigenvalue and two negativeeigenvalues. It then follows from (3.1) that for superstrata (with two anti-self dual tensors) oneshould take ε = +1, while for ε = −
1, the theory has two self-dual tensors and hence the upliftformulae should reduce to a truncation of that given in [1]. Here, we give the full ansatz for the non-linear KK reduction of the six-dimensional theory (3.5)on an S , which gives the three-dimensional gauged supergravity discussed in Section 2, withthree-dimensional metric g µν (coordinates x µ ), 14 scalars which consist of the four scalars χ A and the 10 scalars parametrizing the symmetric matrix m AB , and the six three-dimensionalgauge fields parametrized by the antisymmetric e A ABµ . This reduction ansatz follows from asimple adjustment of the ansatz considered in [1] for six-dimensional N = (2 ,
0) supergravity.
The six-dimensional metric ansatz is: ds = (det m AB ) − / ∆ / ds + g − (det m AB ) / ∆ − / m AB D µ A D µ B , (3.6)where we have made the convenient re-definition of the three-dimensional gauge coupling: g = 2 γ . (3.7)We have also defined: ∆ = m AB µ A µ B . (3.8) Our conventions for the six-dimensional Hodge dual are given explicitly in (3.1). While never explicitlymentioned in [1], their convention for Hodge duals is such that their self-duality relation receives a relative minussign compared to ours in (3.1). µ A , on R are required satisfy µ A µ A = 1, so as to define aunit S . Their gauge-covariant derivatives are D µ A = dµ A − g ˜ A AB µ B ; see Appendix A formore details. Note that the metric ansatz only depends on the scalar matrix m AB (and itsinverse m AB ) and not on the scalars χ A . The six-dimensional scalars are given by the simpleexpressions: e −√ ϕ = ∆ , X = χ A µ A . (3.9) The expressions for the three-forms are quite unwieldy. It is easiest to give the two-form poten-tials B ˆ I , related to the three-forms in the usual way: G ˆ I = dB ˆ I . (3.10)The three-forms G ˆ I and its two-form potentials B ˆ I can be decomposed as: G ˆ I = 13! G ˆ Iijk D y i ∧ D y j ∧ D y k + 12 G ˆ Iijµ D y i ∧ D y j ∧ dx µ (3.11)+ 12 G ˆ Iiµν D y i ∧ dx µ ∧ dx ν + 13! G ˆ Iµνρ dx µ ∧ dx ν ∧ dx ρ ,B ˆ I = 12 B ˆ Iij D y i ∧ D y j + B ˆ Iiµ D y i ∧ dx µ + 12 B ˆ Iµν dx µ ∧ dx ν . (3.12)We will only give expressions for B ˆ Iij and B ˆ Iiµ , which unambiguously determine the compo-nents G ˆ Iijk , G ˆ Iijµ of the three-forms; the other components G ˆ Iiµν , G ˆ Iµνρ (and thus also, by integra-tion, B ˆ Iµν ) are then determined by the self-duality relation (3.1). The ansatze for B ˆ Iij is (usingthe round sphere quantities defined in Appendix A): B ij = (cid:18) − g (cid:19) (cid:18) − ω ijk ˚ ζ k + 12˚ ω ijk ˚ g kl ∆ ∂ l (cid:2) ∆ − (cid:3)(cid:19) , (3.13) B ij = (cid:18) − g (cid:19) (cid:18) ε g − α ˚ ω ijk ˚ ζ k + 14˚ ω ijk ˚ g kl ∆ ∂ l (cid:2) ∆ − X (cid:3)(cid:19) , (3.14) B ij = − √ g ! ω ijk ˚ g kl ∆ / ∂ l (cid:16) ∆ − / X (cid:17) . (3.15)while the ansatze for the components B ˆ Iiµ is: B iµ = (cid:18) − g (cid:19) ∂ i µ A [2 g ] A ABµ (cid:16) µ B − ζ k ∂ k µ B (cid:17) , (3.16) B iµ = (cid:18) − g (cid:19) ∂ i µ A [2 g ] ε (cid:16) − (cid:2) A Aµ B − g − αA ABµ (cid:3) µ B − g − αA ABµ h µ B − ζ k ∂ k µ B i(cid:17) , (3.17) B iµ = 0 . (3.18) Our six-dimensional Hodge dual conventions are given explicitly in (3.1); for completeness, note that we takethe six-dimensional Levi-Civita tensor to decompose as ǫ µνρijk = + ǫ µνρ ǫ ijk .
17n (3.17), the auxiliary gauge field A AB features, but can be integrated out in favor of thefundamental fields in three-dimensions using (2.39), (2.20), and (2.43): g A Aµ B − αA ABµ = − ε ǫ ABCD Y CDµ + 18 ǫ ABCD ǫ νρµ m CC ′ m DD ′ F C ′ D ′ νρ . (3.19)In Section 3.4, we will give explicit formulae for the entire three-forms G ˆ I in a specific sub-sectorrelevant for the (1 , , n ) superstrata.The complete reduction ansatz is thus given by the metric ansatz (3.6), the scalar ansatz(3.9), and the two-form potential ansatze (3.13)-(3.18). Note that there are two constant pa-rameters g and α in the uplift; these (or more precisely, their absolute value) can essentially bechosen at will, as there are two rescalings that one can perform on any six-dimensional solutionwhich correspond to rescaling a three-dimensional theory; we discuss these in Appendix B. Forexample, as we will do in Section 4.2, a natural choice would be to choose g such that m AB = 1lat an asymptotic AdS boundary; then g − is identified with the (asymptotic) S radius in thesix-dimensional solution. The sign of α can be changed by changing ε , as discussed in Section2.7.Our reduction presented here is a simple modification and extension of the N = (2 , S reduction ansatz in [1]; to match their reduction ansatz, we need to take ε = − SO (4)vector of three-forms G α in [1] truncates to our three-form G as G α ∼ δ α, G , and accordinglyfor the three-dimensional scalars. Appendix C contains the explicit matching of our resultsto those of in [1]. (This matching also involves some minor corrections to the uplift formulaepresented [1].) As we noted earlier, the consistent truncation we are interested in is closely related to that ofthe N = (2 ,
0) theory reduced on S . Indeed, our observations in Section 3.1 mean that ourconsistent truncation is essentially guaranteed. However, we still need to establish our upliftformulae and ensure that we have all the details correct. Our tests will also provide extensiveand rigorous testing of the entire consistent truncation more broadly.The first test will be to reduce the theory to a U (1) truncation. Specifically, by imposing thatthe fields are invariant under a particular U (1), we truncate the theory from an SO (4) gaugedtheory to a U (1) gauge theory. This is presented in Section 3.4. For this reduced system weexplicitly checked that the three-dimensional equation of motion (following from the truncatedthree-dimensional Lagrangian (3.26)), together with our uplift ansatz to six-dimensions, implythe six-dimensional three-form Bianchi identities (3.2) and self-duality relations (3.1), as wellas the six-dimensional scalar equations of motion (i.e. the equations of motion for X, ϕ comingfrom (3.5)).The second test involved constructing a new family of six-dimensional solutions: the (1 , m, n )superstrata, which depend on two independent, arbitrary holomorphic functions of one variable.We present these new six-dimensional solutions in Section 4 (and Appendix D) and show thatthey precisely conform to our reduction ansatz. We then extract the three-dimensional data inSection 4.2 and use this as a detailed test of the three-dimensional equations of motion.18inally, in Section 4.3 we examine the overlap of our two tests by looking at the six-dimensional (1 , , n ) superstrata considered as part of the U (1) truncation of Section 3.4.Needless to say, our uplift formulae, and the three-dimensional action, pass all of these tests.More to the point, these tests provide multiple, independent cross checks of all the functionalforms and their coefficients in all of our uplift formulae. In particular, in the uplift we havethoroughly tested all the signs and numerical factors, as well as the appearances of α , g = 2 γ and ε , which correspond precisely to the parameters of the three-dimensional theory. U (1) truncation In this section, we focus on a consistent truncation of the general reduction given in Section 3.3.This truncation is most simply defined by restricting to the fields that are invariant under the O (2) = U (1) subgroup of the SO (4) gauge group that rotates the gauge indices A = 3 , , , n ) superstrata sit, and providesan explicit, more approachable example of the complicated reduction formulae of Section 3.3.There is an analogous truncation that restricts to the sector that is invariant under rotationsthat map the gauge indices A = 1 , , , n ) superstrata. Here we will focus on the first truncation.We will use the explicit coordinates ( θ, ϕ , ϕ ) on the S , see (A.6). The O (2) invariantgauge fields are simply the U (1) Cartan sub-sector: A ABµ = − A ϕ µ × × A ϕ µ ! , A ϕ i µ = A ϕ i µ − A ϕ i µ ! , (3.20)which have been parametrized such that the resulting gauge-covariant coordinates are given by(see (A.2) or (A.4)): D θ = dθ, D ϕ i = dϕ i + g A ϕ i . (3.21)Invariance under O (2) means that we keep χ , χ but set: χ = χ = 0 . (3.22)It also truncates the 10 scalars of m AB to 4 scalars ξ , ξ , ξ , ξ defined as follows: m AB = e − ξ R × × e − ξ × ! , R = exp ξ sin ξ cos ξ cos ξ − sin ξ !! . (3.23)Note that the gauge covariant derivatives on the scalars are given by: D µ ξ , , = ∂ µ ξ , , , D µ ξ = ∂ µ ξ + 2 g A ϕ µ (3.24) D µ χ = ∂ µ χ + g A ϕ µ χ , D µ χ = ∂ µ χ − g A ϕ µ χ . (3.25)Thus, the three-dimensional fields in this truncation are the metric g µν , the two (Abelian)gauge fields A ϕ i µ , and the six scalars ξ , ξ , ξ , ξ , χ , χ . The three-dimensional Lagrangian is19iven by the appropriate truncation of the full three-dimensional Lagrangian (2.40) and can bewritten explicitly as:4 L D,U (1) = R −
12 ( ∂ µ ξ ) −
12 ( ∂ µ ξ ) −
12 ( ∂ µ ξ ) −
12 sinh ξ ( D µ ξ ) (3.26) − e − ξ F ϕ µν F ϕ ,µν − e − ξ F ϕ µν F ϕ ,µν − e ξ (cid:0) cosh ξ (cid:2) ( D µ χ ) + ( D µ χ ) (cid:3) − sinh ξ (cid:2) sin ξ (cid:0) ( D µ χ ) − ( D µ χ ) (cid:1) + 2 cos ξ D µ χ D µ χ (cid:3)(cid:1) + e − ǫ µνρ (cid:18) αA ϕ µ F ϕ νρ + 14 ε F ϕ µν ( χ D ρ χ − χ D ρ χ ) (cid:19) − V,V = − g e ξ (cid:16) e ξ cosh ξ − e ξ sinh ξ (cid:17) + g e ξ + ξ " e ξ (cid:18) ε χ + 12 ε χ + 4 g − α (cid:19) + cosh ξ (cid:0) χ + χ (cid:1) + sinh ξ (cid:0) ( χ − χ ) sin ξ + 2 χ χ cos ξ (cid:1)(cid:3) Note that simply F ϕ i = dA ϕ i .We have checked that the three-dimensional equations of motion following from (3.26) andthe reduction ansatz given in Section 3.3 imply the six-dimensional three-form Bianchi identitiesand self-duality relations, as well as the six-dimensional scalar equations of motion.The (1 , , n ) superstrata solution sits in this truncation (see below in Section 4.3). It is asolution of (3.26) with ξ , = 0. Although one should note that setting these scalar fields to 0does not give a consistent truncation of (3.26), many of the reduction formulae of Section 3.3simplify considerably when these scalars vanish. First of all, we have:∆ | U (1) ,ξ , =0 = m AB µ A µ B = e − ξ sin θ + e − ξ cos θ, X = sin θ ( χ sin ϕ + χ cos ϕ ) . (3.27)The six-dimensional metric ansatz simplifies to: ds (cid:12)(cid:12) U (1) ,ξ , =0 = e ξ + ξ ∆ / ds + g − h ∆ / dθ + ∆ − / e − ξ sin θ D ϕ + ∆ − / e − ξ cos θ D ϕ i . (3.28)In fact, it was this simple metric structure that led to the original conjecture [17] that the(1 , , n ) superstrata should be part of a consistent truncation.Finally, we can also explicitly calculate all components of the three-forms, including thosedetermined by self-duality, which we give here (for ξ , = 0) in form notation: G (cid:12)(cid:12) U (1) ,ξ , =0 = 2 g − ∆ − e − ξ − ξ sin θ cos θdθ ∧ D ϕ ∧ D ϕ (3.29)+ g − ∆ − e − ξ − ξ sin θ cos θ ( dξ − dξ ) ∧ D ϕ ∧ D ϕ − g − ∆ − (cid:16) e − ξ sin θF ϕ ∧ D ϕ + e − ξ cos θF ϕ ∧ D ϕ (cid:17) − e ξ +2 ξ ε (cid:0) ε α + g ( χ + χ ) (cid:1) vol , Note that all Hodge stars in (3.29)-(3.31) refer to three-dimensional Hodge stars with metric ds ; and vol = ∗ G (cid:12)(cid:12) U (1) ,ξ , =0 = − g − (cid:18) ε αg − + 12 ( χ + χ ) (cid:19) sin θ cos θdθ ∧ D ϕ ∧ D ϕ (3.30)+ g − e − ξ − ξ X (cid:16) ∆ − + e ξ ∆ − (cid:17) sin θ cos θdθ ∧ D ϕ ∧ D ϕ + g − ε sin θ cos θ ( − e − ξ ∗ F ϕ ∧ dθ ∧ D ϕ + e − ξ ∗ F ϕ ∧ dθ ∧ D ϕ )+ g − X cos θ (cos ϕ Dχ − sin ϕ Dχ ) ∧ dθ ∧ D ϕ + 12 g − X ∆ − e − ξ − ξ sin θ cos θ ( dξ − dξ ) ∧ D ϕ ∧ D ϕ − g − Xe − ξ ∆ − sin θ cos θ (sin ϕ Dχ + cos ϕ Dχ ) ∧ D ϕ ∧ D ϕ − g − ε cos θ (cid:16) sin θ ( ∗ dξ − ∗ dξ ) + e ξ X (sin ϕ ∗ Dχ + cos ϕ ∗ Dχ ) (cid:17) ∧ dθ − g − X ∆ − ( e − ξ cos θF ϕ ∧ D ϕ + e − ξ sin θF ϕ ∧ D ϕ )+ g − ε e ξ X sin θ (sin ϕ ∗ Dχ − cos ϕ ∗ Dχ ) ∧ D ϕ + ε (cid:18) e ξ + ξ g − e ξ + ξ X (cid:20) g + e ξ (cid:18) ε α + 14 g ( χ + χ ) (cid:19)(cid:21)(cid:19) vol , √ G (cid:12)(cid:12)(cid:12) U (1) ,ξ , =0 = g − ∆ − Xe − ξ − ξ (2 + e ξ ∆) sin θ cos θdθ ∧ D ϕ ∧ D ϕ (3.31)+ g − cos θ (cos ϕ Dχ − sin ϕ Dχ ) ∧ dθ ∧ D ϕ + g − ∆ − h e − ξ − ξ sin θ cos θX ( dξ − dξ ) − sin θ cos θ ∆ e − ξ (sin ϕ Dχ + cos ϕ Dχ ) i ∧ D ϕ ∧ D ϕ − g − ε e ξ cos θ (sin ϕ ∗ Dχ + cos ϕ ∗ Dχ ) ∧ dθ + g − ε e ξ sin θ (sin ϕ ∗ Dχ − cos ϕ ∗ Dχ ) ∧ D ϕ − g − ∆ − X (cid:16) e − ξ cos θF ϕ ∧ D ϕ + e − ξ sin θF ϕ ∧ D ϕ (cid:17) − ε e ξ + ξ X (cid:16) g + e ξ (8 ε α + g ( χ + χ )) (cid:17) vol . To further test our uplift formulae (as well as develop the theory of superstrata), we haveconstructed a novel family of multi-mode superstrata, which are solutions of the six-dimensionaltheory of 3.2.1. We have also verified that they conform to the uplift formula of Section 3.3,and that they give a solution of the three-dimensional theory given by the action (2.40). In thelanguage of [7], this family is produced by superimposing the (1 , m, n ) single-mode superstratawith m ∈ { , } and n ∈ Z + . Since these are the maximal ranges allowed for m and n , werefer to this family of solutions as the (1 , m, n ) multi-mode family.Appendix D contains the full system of six-dimensional BPS equations used to construct thesesolutions, the solutions themselves, along with the regularity and asymptotic charge analysis.Here we give the truncated three-dimensional data, which solve the equations of motion for the These modes are restricted by regularity or equivalently CFT considerations, see [15] for a discussion. , , n ) sub-family, which fitsin the simpler U (1) sub-sector of the six-dimensional reduction given in Section 3.4. We use the S coordinates ( θ, ϕ , ϕ ), with metric (A.7). The S is fibred over a “deformed” AdS , which we parametrize by ( u, v, r ), where u = 1 √ t − y ) , v = 1 √ t + y ) , (4.1)are light cone coordinates, t is the conventional time (in three dimensions) and y parametrizesthe common D1-D5 circle direction with radius R y .Following [7], we introduce the complex coordinate ξ ≡ r √ r + a e i √ vRy . (4.2)A specific (1 , m, n ) multi-mode superstrata is then fixed by specifying the two holomorphicfunctions: F = ∞ X n =1 b n ξ n and F = ∞ X n =1 d n ξ n , (4.3)where ( b n , d n ) are real numbers. Regularity of the solutions requires the introduction of theconstant c = ∞ X n =1 (cid:0) b n + d n (cid:1) , (4.4)with the constraint: 2 Q Q R y = 2 a + c . See Appendix D.4 for details.The (1 , , n ) multi-mode superstrata are recovered by setting F = 0, and the (1 , , n ) multi-mode superstrata are recovered by setting F = 0. (These two multi-mode sub-families werefirst discussed in [7].) (1 , m, n ) superstrata We use the freedom discussed in Appendix B to rescale the six-dimensional uplift formulae ofSection 3.3 (using Λ = 2 p Q /Q ), and then we choose: α = − ε g , g = ( Q Q ) − / . (4.5)We have chosen these constant so that g − = ( Q Q ) / corresponds to the radius of the S insix-dimensions at the asymptotic AdS × S boundary, as appropriate for a D1-D5-P superstrata.22t is convenient to introduce the quantities: S A = − aR y g p a + r ) (cid:18) iF , F , − ie i √ Ry v F , e i √ Ry v F (cid:19) + c.c. . (4.6)The four scalars, χ A , are then given by χ A = 2 S A , (4.7)and the ten scalars in m AB are: m AB = I − S + S S S − S S S S + S S S + S S S + S S S S − S S S S − S S S S + S S S + S S S + S S S S − S S S + S . , (4.8)The three-dimensional metric takes the form of an R fiber over a conformally rescaled two-dimensional K¨ahler manifold: ds = R y g Ω ds − a g du + dv + √ a R y g A ! , (4.9)where: ds = | dξ | (cid:16) − | ξ | (cid:17) , Ω = 2 R y g (1 − S A S A ) , A = i (cid:18) ξ d ¯ ξ − ¯ ξ dξ − | ξ | (cid:19) . (4.10)This shows that the three-dimensional metric has the form of a non-trivial, warped time-fibration over a non-compact CP . This structure is almost certainly a consequence of super-symmetry and is extremely reminiscent of the structure used to find Gutowski-Reall black holesin AdS [43, 44].Finally, the six vector fields e A ABµ read: e A ABµ dx µ = 1 √ a R y g (cid:0) C η AB + C η AB + C η AB + ¯ C ¯ η AB (cid:1) , (4.11) An interesting perspective can also be gained from introducing the complex combinations: z = S + iS and z = S + iS which simplifies some of the following expressions since: S + S = | z | , S + S = | z | , S S − S S = ℜ { z z } , S S + S S = ℑ { z z } . C = ( S S − S S ) d , (4.12) C = ( S S + S S ) d , (4.13) C = (cid:18) a (cid:19) dv − (cid:0) S + S − S − S (cid:1) d , (4.14)¯ C = − (cid:18) a + 2 r (cid:19) dv + (cid:18) − S A S A (cid:19) d , (4.15) d = 1Ω " a ( du + dv ) + 2 r R y g dv , (4.16)and we have introduced the antisymmetric 4 × η AB = σ x − σ x ! , η AB = − σ z σ z ! , η AB = iσ y iσ y ! , (4.17)¯ η AB = − iσ y iσ y ! , ¯ η AB = − II ! , ¯ η AB = iσ y − iσ y ! . (4.18)Note that η j and ¯ η j generate the commuting SU (2) factors of SO (4) = SU (2) × SU (2). Inparticular, this means that the gauge fields in (4.11) define an SU (2) × U (1) gauge connection.We have explicitly checked that the three-dimensional fields ( ds , χ A , m AB , e A ABµ ) givenby (4.9), (4.7), (4.8), and (4.11) with the three-dimensional constants fixed by (4.5) (and(3.7)) satisfy the three-dimensional equations of motion coming from the three-dimensionalLagrangian (2.40) (these are given explicitly in Appendix E). Note that the orientation of thethree-dimensional manifold is tied to the sign of α as we must choose: e − ǫ uvr = − ε , (4.19)where e ≡ p | det( g µν ) | . U (1) truncations The (1 , m, n ) multi-mode solution simplifies greatly when one sets either F = 0, so that S = S = 0, or F = 0, so that S = S = 0. These are the (1 , , n ) and (1 , , n ) multimode solutionsrespectively, both introduced and analyzed in [7]. In each instance the expansion of e A ABµ in(4.11) simplifies with C = C = 0, implying the e A ABµ define a U (1) × U (1) gauge connection.The (1 , , n ) multi-mode family conforms exactly to the U (1) truncation of Section (3.4).Using the notation of that section, the reduction data ( ds , χ , χ , ξ , ξ , ξ , ξ , A ϕ µ , A ϕ µ ), aregiven by: χ , = 2 S , , (4.20)with S = − iaR y g p a + r ) (cid:0) F − ¯ F (cid:1) and S = − aR y g p a + r ) (cid:0) F + ¯ F (cid:1) . (4.21)24he three dimensional metric, ds , again takes the form (4.9)-(4.10) but with the alteredΩ = 2 R y g (1 − S − S ) . (4.22)The remainder of the scalars read: ξ = ξ = ξ = 0 and e − ξ = 12 R y g Ω . (4.23)While the vector fields reduce to: A ϕ µ dx µ = − a R y g √ du + dv ) , (4.24) A ϕ µ dx µ = √ R y g Ω " a ( du + dv ) + 2 a R y g (cid:0) ( a + r )( S + S ) − a (cid:1) dv . (4.25)As noted earlier, the (1 , , n ) multi-mode family is part of another O (2)-invariant truncation.This has χ = χ = 0 and non-trivial ( χ , χ ). This truncated theory will involve a non-trivialgauge coupling in the ϕ direction rather than the ϕ direction. A priori, one might guessthat the (1 , , n ) and (1 , , n ) families are related by a simple change of coordinates ϕ ↔ ϕ .However, we see from (4.15) that the two solutions will have distinct gauge field expansions,even after re-labeling ϕ ↔ ϕ . This is in agreement with the work of [23], where it was alsoshown that the (1 , , n ) and (1 , , n ) single mode solutions are only equivalent after a non-trivialspectral transformation and reduction to five dimensions. We have shown that the three-dimensional (0 ,
2) gauged SO (4) supergravity described in Sec-tion 2 is a consistent truncation of six-dimensional (1 ,
0) supergravity coupled to two tensormultiplets. We have also shown that this consistent truncation includes the newly-constructedfamily of (1 , m, n ) superstrata, which involve momentum waves encoded in two freely-choosableholomorphic functions of one variable.This raises the question as to whether there are other consistent truncations that mightencode yet more classes of microstate geometries. The answer is almost certainly yes. First,the results of [30] suggest that there may well be consistent truncations that encode higher KKmodes, and even entire towers of such modes. These KK towers include the modes of at leastone tensor gauge field and so it seems likely that this work could be extended to the tensorgauge fields that one needs for superstrata.There are also indications that the five-dimensional geometries that can be obtained fromcompactifications of the (2 , , n ) superstrata [45, 17, 23], may also give rise to consistent trun-cations. These would be gauged supergravity theories in three dimensions obtained from AdS × S compactifications of N = 2 supergravity, coupled to vector multiplets, in five dimensions.It therefore seems that the consistent truncations described here might be the tip of aniceberg: there are almost certainly extensive generalizations of our results.25s described in the introduction, our primary interest in examining these consistent trun-cations is to provide a new tool for the study of microstate geometries. In this paper, we haveshown that the consistent truncation contains large and interesting families of BPS superstrata.We plan to see if the three-dimensional approach will enable us to find some new, broaderfamilies of BPS microstate geometries.One of the remarkable things about the six-dimensional BPS equations is that, after speci-fying a hyper-K¨ahler base, the remaining equations reduce to a linear system [16]. A priori , itis not clear whether this simplification will be manifest in the three-dimensional BPS equations.Indeed, it seems likely that linearity in three dimensions will only emerge if one restricts thegauge fields, e A ABµ , to an Abelian sub-sector. These gauge fields may also need to be locked ontothe scalar fields in some manner. An important question then becomes, to what extent one canunlock all the non-abelian gauge fields, while still being able to solve the BPS system? The endresult may well be intrinsically non-linear. If solutions can still be found, then their uplift to sixdimensions might reveal new hyper-K¨ahler bases, which may give new and interesting microstategeometries. As mentioned in the introduction, there is evidence that such bases should exist,coming from perturbation theory, in both [24, 25].Even more important is the possibility of constructing non-BPS microstate geometries.Through a simple parity flip, one can convert BPS superstata into anti-BPS superstata (onesthat preserve a different, complementary set of supersymmetries). It is therefore possible touse the three-dimensional formulation to study non-BPS configurations that start from a com-bination of BPS and anti-BPS momentum waves. It should be relatively straightforward toset up a three-dimensional initial value problem that should produce such non-BPS microstategeometries as the result of ‘scattering’ BPS and anti-BPS waves. The extent to which this canbe done analytically, or semi-analytically, is unclear, but it will certainly be possible to studythis numerically.In considering the outcome of such an approach to non-BPS solutions, it is important toremember the Faustian bargain of consistent truncations. It is quite possible that the combin-ing of BPS and non-BPS solutions in three dimensions will evolve, at late times, into a singularsolution. As we have seen in many examples of microstate geometries, the appearance of asingularity in supergravity does not invalidate the microstate geometry program, but usuallyindicates that one has suppressed degrees of freedom that are essential to resolving the singu-larity. Thus the appearance of a singularity at late times may simply be the result of limitingthe degrees of freedom to a consistent truncation.Even if singularities do arise in such non-BPS solutions, there will still be invaluable informa-tion to be gleaned from the three-dimensional analysis. One will see the early, time-dependentbehavior and the radiation that comes from the scattering. By using the uplifts one may alsobe able to determine which degrees of freedom will be needed to resolve any singular behaviour.It is also possible that the microstate geometries created in this way will be smooth and robustand provide families of non-BPS microstate geometries for which the holographic dictionary isprecisely known. 26 cknowledgments
NPW would like to thank Henning Samtleben for his patient and careful explanations of manyaspects of his work on gauged supergravity in three dimensions. DRM is supported by the ERCStarting Grant 679278 Emergent-BH. The work of NPW and RAW was supported in part byERC Grant number: 787320 - QBH Structure and by DOE grant DE-SC0011687. RAW is verygrateful to the IPhT of CEA-Saclay for hospitality during this project, his research was alsosupported by a Chateaubriand Fellowship of the Office for Science Technology of the Embassyof France in the United States.
A The Three-sphere
It is convenient to parametrize the unit radius, round S with four restricted Cartesian coordi-nates µ A of R that satisfy µ A µ A = 1, or alternatively with three (unrestricted) coordinates y i .The round, unit-sphere metric in coordinates y i is ˚ g ij , with corresponding completely antisym-metric tensor ˚ ω ijk . Following [1], we also use a vector ˚ ζ i with unit divergence,˚ ∇ i ˚ ζ i = 1 . (A.1)The gauge-covariant derivatives on the sphere are then: D µ A = dµ A − g ˜ A AB µ B , (A.2)where g = 2 γ is the gauge coupling, and we have used the dual gauge fields given in (2.43).We can rewrite this as: D µ A = ∂ i µ A D y i , (A.3)with: D y i = dy i − g K iAB ˜ A AB , (A.4)where we have used the Killing vectors on the sphere: K iAB = ˚ g ij ∂ j µ [ A µ B ] . (A.5)There are many other identities involving the µ A (which we will not explicitly need in thispaper); see, for example, Appendix A of [1].An explicit coordinate basis that can be used is, for example, the standard coordinates y i = ( θ, ϕ , ϕ ) with: µ = sin θ sin ϕ , µ = sin θ cos ϕ , µ = cos θ sin ϕ , µ = cos θ cos ϕ . (A.6)The metric in these coordinates of the unit radius round S is:˚ ds S = ˚ g ij dy i dy j = dθ + sin θdϕ + cos θdϕ , (A.7)so that ˚ ω ijk = (sin θ cos θ ) ǫ ijk , with ǫ = +1 and completely antisymmetric. In these coordi-nates, we can take: ˚ ζ i = (cid:18)
12 tan θ, , (cid:19) . (A.8)27 Six-dimensional and three-dimensional rescalings
We wish to point out two rescalings of the six-dimensional fields which have a counterpart as arescaling of three-dimensional fields, through the uplift formulae in Section 3.3.The first rescaling is: e √ ϕ → Λ − e √ ϕ , X → Λ X, g (6 D )ˆ µ ˆ ν → Λ g (6 D )ˆ µ ˆ ν , (B.1) G → G , G → Λ G , G → Λ G , (B.2)which corresponds to the three-dimensional rescaling: m AB → Λ m AB , χ A → Λ χ A , (B.3) α → Λ α, g (3 D ) µν → Λ g (3 D ) µν . (B.4)Under this scaling, the six-dimensional Lagrangian (3.5), resp. three-dimensional action(2.40), scales as ˆ e L D → Λ ˆ e L D , resp. e L D → Λ e L D (with ˆ e = q − det g (6 D )ˆ µ ˆ ν and e = q − det g (3 D ) µν ).The second rescaling is: e √ ϕ → Λ − e √ ϕ , X → Λ X, g (6 D )ˆ µ ˆ ν → g (6 D )ˆ µ ˆ ν , (B.5) G → Λ − G , G → Λ G , G → G , (B.6)which has the three-dimensional counterpart: m AB → Λ m AB , χ A → Λ χ A , (B.7) α → Λ / α, g (3 D ) µν → Λ g (3 D ) µν , (B.8) g → Λ / g , A ABµ → Λ − / A ABµ . (B.9)Note that the rescaling of g implies the same rescaling of γ through (3.7). The six-dimensionalaction (3.5) is invariant under this scaling, ˆ e L D → ˆ e L D , while the three-dimensional action(2.40) rescales as e L D → Λ / e L D .A combination of both of these scalings can be used to rescale the two constants | α | and g to any value in the reduced three-dimensional theory. C Matching with the conventions of [1]
Our goal here is to provide a map between our conventions and those of [1].
C.1 Consistency of gauge field actions, Chern-Simons term and conventions
There is some tension between our formulation of the gauge action, and that of [1]. Here wediscuss the differences in detail and describe why we have provided a consistent set of conventions.28ssentially there are four signs that must be correctly correlated: (i) The sign of the rep-resentation matrices that define the minimal couplings, (ii) the sign of the structure constants,(iii) the sign of the A ∧ A term in the field strength, and (iv) the sign of the A ∧ A ∧ A term inthe CS action.We have defined the covariant derivative by (2.12): b D µ X ¯ P ≡ ∂ µ X ¯ P + A µ ¯ K ¯ L Θ ¯ K ¯ L, ¯ M ¯ N ( T ¯ M ¯ N ) ¯ P ¯ Q ( X ¯ Q ) , (C.1)and have chosen the representation matrices as so as to obtain (2.13), which means we havetaken: ( T ¯ M ¯ N ) ¯ P ¯ Q = δ ¯ N ¯ P η ¯ M ¯ Q − δ ¯ M ¯ P η ¯ N ¯ Q . (C.2)This then led to the covariant derivatives given in (2.21) and ultimately to the covariant deriva-tives in (2.26). D µ X A = ∂ µ X A − γ e A µAB X B , D µ X A = ∂ µ X A − γ e A µAB X B , D µ X = ∂ µ X . (C.3)We note that it follows from this that the SO (4) field strength is given by (cid:2) D µ , D ν ] X A ≡ − γ e F µν AB X B = − γ (cid:16) ∂ µ e A νAB − ∂ ν e A µAB − γ (cid:0) e A µAC e A ν CB − e A νAC e A µCB (cid:1) (cid:17) X B , (C.4)from which one obtains F µν AB = ǫ ABCD e F µν CD = 2 (cid:0) ∂ [ ν A ρ ] AB − γ A [ νC [ A e A ρ ] B ] C (cid:1) . (C.5)It is necessary for consistency, that this is precisely the field strength given in (2.34). The latterexpression was obtained from the Chern-Simons action after integrating out the T gauge fields, B µAB . The important message here is that our explicit expressions for the gauge covariantderivatives and gauge actions are consistent with one another.The simplest way to obtain the Chern-Simons action is to work with F ∧ F in higher dimen-sions and write it as d ( A ∧ F + A ∧ A ∧ A ), and then the term in parentheses is the action weseek. This leads to F a = dA a + f abc A b ∧ A c , L CS = A a ∧ dA a + f abc A a ∧ A b ∧ A c , (C.6)from which it follows that the Chern-Simons we seek is given by (2.32): L CS = 14 ε µνρ A µ ¯ K ¯ L Θ ¯ K ¯ L, ¯ M ¯ N (cid:16) ∂ ν A ρ ¯ M ¯ N + 13 f ¯ M ¯ N, ¯ P ¯ Q ¯ R ¯ S Θ ¯ P ¯ Q, ¯ U ¯ V A ν ¯ U ¯ V A ρ ¯ R ¯ S (cid:17) , (C.7)In passing from the general expression (C.6) to (C.7), one should remember that the role of theembedding tensor, Θ, is simply that of a projector from the large algebra, SO (4 , SO (4) × T and thus (C.6) is the appropriate expression on the gauge Liealgebra after projection. 29he corresponding expression in [1] (equation (2.6)) is “non-canonical” in that the sign ofthe A ∧ A ∧ A term is reversed relative to our “canonical choice.” Earlier references, like [46, 28],have the canonical form of the Cherns-Simons term, (C.7).One should also note that our choice of representation matrices, (C.2), leads to the oppositesign of the commutators and structure constants, (2.14) and (2.15), when compared to [1]. Thismeans that our final Chern-Simons action actually matches that of [1]. However, our minimalcouplings and representation matrices have the opposite sign to those of [1] and thus we believethere is a potential inconsistency in the complete action of [1].We have used the “canonical” form of the Chern-Simons terms, together with a consistentchoice the generators (C.2) and field strength. The equations of motion are sensitive to allof these sign choices and the fact that the (1 , m, n ) superstrata solve the resulting equationsof motion, with all of these convention, give us further confidence that our conventions areconsistent. C.2 Matching the uplift formulae
We start by noting that our three-dimensional formulation matches that of [1] if one sets γ = 1 ⇔ g = 2 . (C.8)The parameter α is the same in both sources.Turning to the six-dimensional theory, our scalar matrix is M ˆ I ˆ J = 12 e √ ϕ X √ XX e − √ ϕ (2 + e √ ϕ X ) √ (2 e −√ ϕ + X ) X √ X √ (2 e −√ ϕ + X ) X e −√ ϕ + X ) , (C.9)where we have raised the second index using our SO (1 ,
2) metric: η ˆ I ˆ J = − . (C.10)One should also recall (3.8) and (3.9): e −√ ϕ = ∆ = m AB µ A µ B . (C.11)The corresponding objects in [1] are f M ab =18
4∆ + 4 X + ∆ − (2 + X ) − − ∆(2 + ∆ − X ) − √ − (2 + X )) X − − ∆(2 + ∆ − X )
4∆ + 4 X + ∆ − (2 − X ) √ − ∆ − (2 − X )) X − √ − (2 + X )) X √ − ∆ − (2 − X )) X − X , (C.12)30nd ˜ η ab = − − , (C.13)where the indices a, b, . . . take the values 0 , ¯0 , ¯1. To compare with the results of [1], one shouldnote that [1] uses two expressions: ∆ and ˜∆. We re-label the ∆ of [1] as ˆ∆ here; the relationsbetween these quantities and our ∆ are:ˆ∆ = (cid:0) det( m AB ) (cid:1) ∆ − , ˜∆ = ∆ − . (C.14)Define the matrix P = − √ −√ − √ √ √ , (C.15)then one can easily verify that˜ η = P η P t , f M = P M P t , (C.16)Thus P provides a change of basis from our fields to those of [1]. In particular, performing thechange of basis on the gauge potentials (3.13)–(3.15) yields B ij = − √ B ij − √ B ij = 1 √ g h − (cid:0) − εαg − (cid:1) ˚ ω ijk ˚ ζ k − ˚ ω ijk ˚ g kl ∂ l (log ∆) + ˚ ω ijk ˚ g kl ∆ ∂ l (∆ − X ) i , (C.17) B ¯0 ij = − √ B ij + √ B ij = 1 √ g h − (cid:0) εαg − (cid:1) ˚ ω ijk ˚ ζ k − ˚ ω ijk ˚ g kl ∂ l (log ∆) − ˚ ω ijk ˚ g kl ∆ ∂ l (∆ − X ) i , (C.18) B ¯1 ij = 1 √ B ij = − g ˚ ω ijk ˚ g kl ∆ / ∂ l (cid:16) ∆ − / X (cid:17) , (C.19)Similarly, transforming (3.16) – (3.17) yields B µi = − √ B iµ + √ B iµ = − √ g (cid:0) ∂ i µ A (cid:1) h(cid:0) A ABµ − ε A µAB (cid:1) µ B − (cid:0) − εαg − (cid:1) A ABµ (cid:0) ˚ ζ k ∂ k µ B (cid:1)i , (C.20) B ¯0 µi = 12 √ B iµ − √ B iµ = − √ g (cid:0) ∂ i µ A (cid:1) h(cid:0) A ABµ + ε A µAB (cid:1) µ B − (cid:0) εαg − (cid:1) A ABµ (cid:0) ˚ ζ k ∂ k µ B (cid:1)i , (C.21) B ¯1 µi = − √ B iµ = 0 . (C.22)31ote that we have reversed the indices, µi , on the left-hand side to facilitate comparison with [1].We find a perfect match for the B ij components, up to an overall factor of g − , providedthat one uses (C.8) and takes ε = − . (C.23)This choice was anticipated in Sections 2.2 and 3.2.1.Using (C.8) and (C.23) we also find a nearly perfect match for the B µi components, up toan overall factor of − g − .The overall factors of g − in B ij and − g − in B µi are easily fixed using the scalings inAppendix B. Indeed, choosing Λ = Λ − = Λ (C.24)results in the six-dimensional rescaling g (6 D )ˆ µ ˆ ν → Λ g (6 D )ˆ µ ˆ ν , G ˆ I → Λ G ˆ I . (C.25)One can then take Λ = g = 4 to match the overall scale in B ij .This scaling then creates an overall factor of − g in B µi . This can then be compensated bycoordinate changes: x µ = ˜ x µ , y i = − ˜ y i , (C.26)which then rescale B µi by − , while leaving B ij unchanged.There are two discrepancies between our analysis and that of [1] that may be transcriptionerrors in [1]. First, although our expressions for B ij , B ¯0 ij , B ¯1 ij match those in [1] precisely, ourexpressions for B µi and B ¯0 µi match the expressions in [1] for B µi and B µi ¯0 . When 0 and ¯0 arelowered using ˜ η ab (see, C.13), they get a relative minus sign and so they cannot be reconciledsimultaneously.Second, we have, using (C.8): D y i = dy i − g K iAB ˜ A AB = − d ˜ y i − K iAB ˜ A AB , (C.27)whereas, [1] defines D y i = dy i + K iAB ˜ A AB , (C.28)so these expressions do not match. We also note that the combination g A ABµ dx µ is bothcoordinate invariant and invariant under both the rescalings described in Appendix B, and sowe cannot reconcile our expressions for D y i with those of [1].Therefore, up to two minor discrepancies, our results match the expressions in [1]. As weindicated in Section 3.3.3, we have subjected our uplift formulae to rigorous testing in both threeand six dimensions, and have every confidence in our expressions and normalizations. D (1 , m, n ) Superstrata in six dimensions
This appendix summarizes the six-dimensional BPS equations for the D1-D5-P system, and thenovel construction of the (1 , m, n ) multi-mode superstrata family of solutions.32 .1 Six-dimensional BPS equations
All -BPS solutions of six-dimensional, N = (1 ,
0) supergravity, coupled to 2 tensor multiplets,with the same charges as the D1-D5-P system, satisfy a “layered” set of linear equations. Theseequations were first developed in [16] for a single tensor multiplet, and extended to include asecond tensor multiplet in [39]. We follow [45], introducing the equations in an explicitly SO (1 , SO (1 ,
2) indices: ˆ I, ˆ J , ˆ K, · · · ∈ { , , } , with non-zero SO (1 ,
2) metriccomponents (see (3.4)): η = η = 1 and η = − . (D.1)The full six-dimensional geometry, constrained by supersymmetry, may be written as a(1 + 1)-dimensional Lorentzian fiber, parametrized by the light cone coordinates ( u, v ) (see(4.1)), over a four dimensional hyper-K¨ahler base ds ( B ) as: ds = − √P ( dv + β ) (cid:0) du + ω + F ( dv + β ) (cid:1) + √P ds ( B ) . (D.2)The metric data consists of the base B , the functions ( P , F ) and one forms ( β, ω ). The one formsmust have legs only on the base B , while the complete data can have functional dependence onall coordinates except for u . This metric data is fixed by solving the BPS equations, which arewritten in terms of a set of three functions Z ˆ I and three two forms Θ ˆ I . In terms of this data,the three-form fields encoding the multiplets read: G ˆ I = d " − η ˆ I ˆ J Z ˆ J P ( du + ω ) ∧ ( dv + β ) + 12 η ˆ I ˆ J ∗ DZ ˆ J + 12 ( dv + β ) ∧ Θ ˆ I (D.3)where: P = 12 η ˆ I ˆ J Z ˆ I Z ˆ J = Z Z − ( Z ) , (D.4) D is defined to act on forms Φ by: D Φ = d Φ − β ∧ ˙Φ (D.5)where overhead dots denote ∂ v derivatives, ( d , ∗ ) are the exterior derivative and Hodge starwith respect to the four-dimensional hyper-K¨ahler base B , and their non-subscript and hattedcounterparts refer to the full six-dimensional geometry (D.2). These three forms satisfy thetwisted self duality constraint: ˆ ∗ G ˆ I = M ˆ I ˆ J G ˆ J where M ˆ I ˆ J = Z ˆ I Z ˆ J P − η ˆ I ˆ J . (D.6)Solving the BPS equations takes the layered form: • Fix a hyper-K¨ahler base ds ( B ) and choose a β satisfying: dβ = ∗ dβ . (D.7) Note that this corresponds to choosing ε = +1 in (3.1). Also note that our Hodge dual conventions (as givenin (3.1)) imply that there should indeed be two anti-self-dual tensors and one self-dual tensor for the superstrata. Find a set ( Z ˆ I , Θ ˆ I ) that solve the “first layer” : ∗ D ˙ Z ˆ I = η ˆ I ˆ J D Θ ˆ J , D ∗ DZ ˆ I = − η ˆ I ˆ J Θ ˆ J ∧ dβ , Θ ˆ I = ∗ Θ ˆ I . (D.8) • Find ( F , ω ) that solve the “second layer:”(1 + ∗ ) Dω + F dβ = Z ˆ I Θ ˆ I , (D.9) ∗ D ∗ (cid:18) ˙ ω − D F (cid:19) = 14 η ˆ I ˆ J h Z ˆ I Z ˆ J + 2 ˙ Z ˆ I ˙ Z ˆ J − ∗ (cid:16) Θ ˆ I ∧ Θ ˆ J (cid:17)i . (D.10) D.2 The solution
The standard hyper-K¨ahler base used in the construction of six-dimensional superstrata is flat R , which is most conveniently written in spherical bipolar coordinates ( r, θ, ϕ , ϕ ), with metric: ds ( B ) = Σ (cid:18) dr r + a + dθ (cid:19) + ( r + a ) sin θ dϕ + r cos θ dϕ . (D.11)where a is a positive constant and Σ ≡ r + a cos θ . (D.12)In terms of the complex coordinates: χ ≡ a √ r + a sin θ e iϕ , µ = cot θ e i (cid:0) √ vRy − ϕ − ϕ (cid:1) , ξ ≡ r √ r + a e i √ vRy , (D.13)the (1 , m, n ) multi-mode solution can be written in terms of the basic function: F ( χ, µ, ξ ) = χF ( ξ ) + χµF ( ξ ) , (D.14)where F , are holomorphic functions of ξ , with expansions in terms of the real coefficients( c n , d n ): F = ∞ X n =1 b n ξ n and F = ∞ X n =1 d n ξ n . (D.15)We define the auxiliary data: A = χµ (1 + ξ∂ ξ ) F and B = χξ∂ ξ F , (D.16)and self dual formsΩ y = 1 √ (cid:16) − Ω (2) + ir sin θ Ω (1) (cid:17) , Ω z = 1 √ (cid:18) Ω (3) + i (cid:18) r sin θ − Σ r sin θ (cid:19) Ω (1) (cid:19) . (D.17) Significant process was made in solving this layer in general in [25]. For any harmonic functions Φ ˆ I on B , onecan derive from them and a complex structure a self dual two forms Θ ˆ I . If it is known what modulus of B thesetwo form control as a K¨ahler deformation, then the Z ˆ I which solve the first BPS layer together with these Θ ˆ I canbe found directly from the Φ ˆ I . (1) ≡ dr ∧ dθ ( r + a ) cos θ + r sin θ Σ dϕ ∧ dϕ , Ω (2) ≡ rr + a dr ∧ dϕ + tan θ dθ ∧ dϕ , Ω (3) ≡ dr ∧ dϕ r − cot θ dθ ∧ dϕ . (D.18)To solve the BPS equations, first, one fixes: β = a R y √ (cid:0) sin θ dϕ − cos θ dϕ (cid:1) , (D.19)then the solution to the first BPS layer is given by the data: Z = Q Σ + R y Q Σ (cid:0) F + ¯ F (cid:1) ,Z = Q Σ ,Z = R y (cid:0) F + ¯ F (cid:1) , Θ = 0 , Θ = R y Q F ( A Ω y + B Ω z ) + c.c. , Θ = − A Ω y + B Ω z ) + c.c. . (D.20)The solution to the second BPS layer can then be written in the form: F = F ( p ) + c F ( c ) (D.21) ω = 4sin 2 θ ω ( p ) µ dθ + 2 (cid:16) ω (0) χ + ω ( p ) χ + c ω ( c ) χ (cid:17) dϕ + 2 (cid:16) ω (0) δ + ω ( p ) δ (cid:17) dϕ , (D.22)where c is a constant. The “round supertube” part is given by: ω (0) χ = ω (0) δ | µ | = R y | χ | √ − | χ | ) . (D.23)The homogeneous part is given by: F ( c ) = − a and ω ( c ) χ = R y | χ | √ a (1 − | χ | ) . (D.24)Finally, the solution is completed by adding the particular part: F ( p ) = 1 a (cid:16) | F | + | ξ | | F | (cid:17) , ω ( p ) χ = − R y √ a (1 − | χ | ) (cid:0) ¯ χ ¯ F F + χF ¯ F (cid:1) , (D.25) ω ( p ) µ = − iR y | χ | √ a (cid:0) µ ¯ F F − ¯ µF ¯ F (cid:1) , ω ( p ) δ = R y | ξ | √ a (1 − | χ | ) (cid:0) χµF ¯ F + ¯ χ ¯ µ ¯ F F (cid:1) . (D.26) Note that we introduce this constant of integration as c , whereas c (i.e. unsquared) was used in [7]. As wewill see in (D.29), c is naturally a positive number. .3 Tuning the asymptotic geometry Setting F = F = 0 in the (1 , m, n ) solution of the previous section gives the “round supertube”solution, which is globally AdS × S . This solution has F = 0 and ω = ω = a R y √ (cid:0) sin θ dϕ + cos θ dϕ (cid:1) . (D.27)To ensure the (1 , m, n ) solutions have the same asymptotics, one must arrange for ( F , ω ) tohave at most O ( r − ) corrections to the round supertube solution (D.27). We achieve this byfirst defining: F ( ∞ )0 ( v ) ≡ lim | ξ |→∞ F ( ξ ) = lim r →∞ F ( ξ ) and F ( ∞ )1 ( v ) ≡ lim | ξ |→∞ F ( ξ ) = lim r →∞ F ( ξ ) , (D.28)and then fixing c by: c ≡ √ πR Z dv (cid:18)(cid:12)(cid:12)(cid:12) F ( ∞ )0 ( v ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) F ( ∞ )1 ( v ) (cid:12)(cid:12)(cid:12) (cid:19) = b + d , (D.29)where we have implicitly defined b = ∞ X n =1 b n and d = ∞ X n =1 d n . (D.30)Then, we can use a gauge transformation which leaves the six-dimensional BPS equations (D.7)-(D.10) and metric (D.2) invariant: u → u + f ( v, r, θ, ϕ , ϕ ) ⇐⇒ ω → ω − d f + ˙ f β , F → F − f , (D.31)with gauge parameter chosen as: f ( v ) ≡ a Z v dv ′ (cid:18)(cid:12)(cid:12)(cid:12) F ( ∞ )0 ( v ′ ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) F ( ∞ )1 ( v ′ ) (cid:12)(cid:12)(cid:12) − c (cid:19) , (D.32)which brings the ( F , ω ) for the (1 , m, n ) family to the form F = − r d + X n =1 n ( b n + d n ) + oscillating terms ! + O ( r − ) , (D.33) ω = ω + R y d √ r (cid:0) sin θ dϕ + cos θ dϕ + oscillating terms (cid:1) + O ( r − ) . (D.34)This gauge-transformed geometry is now asymptotically the same as the round supertube, i.e. AdS × S . Note that the relation (D.29) is crucial to make the gauge parameter (D.32) awell-defined, periodic function of v ; without imposing (D.29) it is not possible to retrieve thecorrect asymptotics. 36 .4 Regularity and CTC analysis There are four distinct ways in which the six-dimensional metric (D.2), for the (1 , m, n ) solutionof the previous section, which also takes the form (3.6), may fail to be regular: • The metric is singular where the data ( β, ω, F ) are singular, at the locus:Σ = 0 . (D.35) • The warp factors (∆ − det m AB ) ± / are singular. • The sphere deformations e A ABµ of (4.11) are singular. • The ds metric (4.9)-(4.10) possesses a conical singularity at r = 0, where the y -circlepinches off. • ds possesses closed time-like curves (CTCs).Upon expanding and analyzing the metric along the locus (D.35), the only potentially sin-gular part was found to be the dϕ coefficient. Setting r = aǫ and θ = π − ǫ , (D.36)and expanding in powers of ǫ , this term reads:1 ǫ Q Q − (2 a + c ) R y q Q Q − | F | R y dϕ + O ( ǫ ) . (D.37)To remove this singularity one must tune: Q Q R y = 1 g R y = a + c . (D.38)Now consider det m AB = (1 − S A S A ) = (cid:16) − | z | − | z | (cid:17) , (D.39)where z = S + iS and z = S + iS . (D.40)Since m AB = ( m AB ) − appears in the kinetic term of the three-dimensional Lagrangian (2.40),its solutions must necessarily bound det m AB away from zero . Since lim r →∞ S A = 0 weconclude that: 0 < (det m AB ) / = 1 − | z | − | z | , (D.41) This argument is plausible, rather than providing a strict proof. The scalar action would become infinitewherever det m AB = 0, so the minimization procedure should ensure solutions avoid this condition. , m, n ) solution. Now one can alsocalculate that ∆ = 1 − | ˜ z + ˜ z | , (D.42)where ˜ z = ( S − iS ) sin θ e − iϕ and ˜ z = − ( S + iS ) cos θ e iϕ . (D.43)The triangle inequality and (D.41) then imply that0 < ∆ . (D.44)Hence the warp factors (∆ − det m AB ) ± / are regular.In passing, looking at the form of Ω in (4.10), it also follows that:0 < Ω . (D.45)Hence the sphere deformations e A ABµ of (4.11) are clearly regular by inspection.Setting ρ = r/a , the three dimensional metric can be written as: ds = 1 g (cid:20) dρ ρ − g a R y (1 + ρ ) dt + ρ R y (cid:0) dy + (1 − g a R y ) dt (cid:1) (cid:21) − g (cid:16) | F | + | F | (cid:17) " ρ ( dt + dy ) + R y dρ (1 + ρ ) . (D.46)This form of the metric makes it clear that there is no conical singularity when the y -circlepinches off at ρ = 0.Proving there are no CTCs in (D.46), when suitably regularized by (D.38), and tuned tobe asymptotically AdS by (D.29), is a delicate business. A proof for the (1 , , n ) and (1 , , n )families appears in [7]. It relies on properties of | F , | following from the analyticity of F , ,which do not easily generalize to the sum | F | + | F | , as is required for the (1 , m, n ) family.Although we do not have a proof, we expect ds to be free of CTCs when tuned with (D.29)and (D.38) for the (1 , m, n ) family. This expectation is based on examining many examples withexplicit expansions of F , , as well as the fact that the holographic duals should be well definedCFT states. Intuitively, one can think of the second line of (D.46) as a “perturbation” of aregular AdS seed. Upon fixing the CFT charges Q , , the magnitude of | F , | are restrictedby (D.29) and (D.38), and so the negative contribution coming from the “perturbation” issufficiently controlled so as to avoid CTCs. D.5 Conserved charges
A detailed analysis of computing the conserved charges for the six-dimensional superstrataappears in [7], where the explicit calculations for the (1 , , n ) and (1 , , n ) multi-mode familiesare also given. Here we give a short summary of the procedure, which are closely analogous to38he individual (1 , , n ) and (1 , , n ) family analysis, and the result. The analysis requires c tobe tuned as in (D.29), so that the geometry is asymptotic to AdS × S .The D1-D5-P system possesses five conserved charges: the net charge of each type of brane Q , , the momentum in the common D1-D5 direction Q P , and the two angular momenta J L,R .The brane charges Q , can be simply read off from the r − coefficient of Z , in (D.20), whenexpanded about r → ∞ .Using the solution as presented in the gauge of (D.33)-(D.34), with c fixed by (D.29), theremaining charges can be read off from the expansions: β + β + ω + ω = √ r (cid:2) ( J R − J L cos 2 θ ) + oscillating terms (cid:3) + O ( r − ) (D.47)where β , ω and β , ω are the components of β and ω along dϕ and dϕ respectively, and F = − r (cid:0) Q P + oscillating terms (cid:1) + O ( r − ) . (D.48)For the (1 , m, n ) multi-mode solution this procedure gives (using (D.30)): J L = a R y , J R = R y (cid:0) a + d (cid:1) , Q P = 12 " d + ∞ X n =1 n (cid:0) b n + d n (cid:1) . (D.49) D.6 Uplifting the six-dimensional scalars to ten dimensions
The parametrization of the six-dimensional scalars ϕ, X that we used in the uplift formulae inSection 3.3 is convenient from the point of view of the six-dimensional reduction, as ϕ , resp. X ,only depend on m AB , resp. χ A . However, for the D1-D5 system, they require some rearrangingto consider the uplift to ten dimensions; in particular, the ten-dimensional dilaton is not givenby ϕ . With the following rearrangement of the scalars, introducing an arbitrary constant ˜ Z : e φ (10D) = e √ ϕ (cid:16) e −√ ϕ + X (cid:17) Z , C (10D)(0) = −√ Z X e −√ ϕ + X , (D.50)the six-dimensional scalar kinetic action becomes: L D, scal.kin = −
12 ( ∂ϕ ) − e √ ϕ ( ∂X ) = − ( ∂φ (10D) ) − e φ (10D) ( ∂C (10D)(0) ) . (D.51)Then, for the superstrata uplift to ten dimensions (using the conventions of, for example, Ap-pendix B of [42]), φ (10D) can be identified with the ten-dimensional dilaton after uplifting on a T while C (10D)(0) is the ten-dimensional axion. Setting ˜ Z = Z ( g Σ) then gives the traditionalform for these scalars: e φ (10D) = Z P , C (10D)(0) = Z Z , (D.52)whereas the six-dimensional fields used in Section 3.3 were simply: e −√ ϕ = ( g Σ) P , X = −√ g Σ) Z . (D.53)39 Three-dimensional equations of motion
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