AdS3: the NHEK generation

HJE AdS3: the Iosif Bena 0 1 3 6 7 8 Lucien Heurtier 0 1 3 4 5 7 8 Andrea Puhm 0 1 2 3 7 8 0 Boulevard du Triomphe , CP225, 1050 Brussels , Belgium 1 F-91128 Palaiseau , France 2 Department of Physics, University of California 3 F-91191 Gif sur Yvette , France 4 Centre de Physique Theorique, Ecole Polytechnique, CNRS, Universite Paris-Saclay 5 Service de Physique Theorique, Universite Libre de Bruxelles 6 Institut de Physique Theorique, CEA Saclay 7 ve-dimensional near-horizon extremal Kerr 8 Santa Barbara , CA 93106 , U.S.A It was argued in [1] that the (NHEK) geometry can be embedded in String Theory as the infrared region of an innite family of non-supersymmetric geometries that have D1, D5, momentum and KK monopole charges. We show that there exists a method to embed these geometries into asymptotically-AdS3 S3=ZN solutions, and hence to obtain in nite families of ows whose infrared is NHEK. This indicates that the CFT dual to the NHEK geometry is the IR AdS-CFT Correspondence; Black Holes in String Theory; D-branes - xed point of a Renormalization Group ow from a known local UV CFT and opens the door to its explicit construction. 2.1 2.2 2.3 3.1 3.2 3.3 1 Introduction 2 In nite families of NHEK embeddings in String Theory Almost-BPS D1-D5-P-KK solutions Non-BPS D1-D5-P-KK solutions with a NHEK infrared Towards the full explicit form of the R-R elds 3 An AdS3 throat with a NHEK Solutions with NHEK infrared Hunting for AdS asymptotics Constraints for AdS asymptotics 3.3.1 3.3.2 The Kerr-CFT conjecture [2] relates the near-horizon geometry of an extremal Kerr black hole [3] to a putative 1+1 dimensional conformal eld theory whose central charges are given by the angular momenta of this black hole. This connection allows one to count microscopically the entropy of extremal Kerr black holes, and its discovery is an important step in extending the powerful machinery of string theory to analyze non-supersymmetric black holes. Unfortunately, besides the central charge, very little is known about the CFT dual to the Near-Horizon Extremal Kerr (NHEK) geometry. In particular, it is not known whether this CFT can be realized as the infrared xed point of a Renormalization Group (RG) ow from a known UV theory, or as a low-energy theory on a system of strings and branes. In order to achieve such a construction it is important to embed the NHEK geometry in String Theory, and to look for a system of branes that, when gravity is turned on, gives rise to a NHEK geometry. The rst attempt at such an embedding was made by the authors of [4, 5], who constructed a solution with D1, D5 and Taub-NUT charges, that has a vedimensional NHEK geometry in the infrared. However, this solution has xed moduli, and hence does not allow one to search for a microscopic theory that ows in the infrared to the CFT dual to NHEK. { 1 { In [ 1 ], it was shown that one can embed the ve-dimensional NHEK geometry in a very large family of supergravity solutions, parameterized by several continuous parameters, and that moreover one can obtain multicenter solutions where the geometry near one of the centers is NHEK. All these solutions belong to a class of extremal non-supersymmetric solutions that can be obtained by performing a duality sequence known as generalized spectral ow [6] on the well-known almost-BPS solutions [7, 8].1 The existence of this very large family of solutions that have a NHEK region in the infrared raises the hope that one may be able to nd a ow from a UV that is AdS to the NHEK geometry in the infrared. This would imply that the UV CFT is a \nice" local CFT, with well-understood operators, etc. One could then go ahead and investigate this UV CFT and nd which operators procedures to embed the in nite families of multicenter solutions with NHEK regions [ 1 ] into asymptotically-AdS3 solutions. The rst procedure is to use the explicit form of the solutions [ 1 ] and to investigate various limits of the parameters that control the UV of these solutions in order to produce an AdS factor in the metric. The second is to make clever use of the fact that the only di erence between the asymptotics of BPS and almost-BPS solutions is the sign of one of the electric elds [7], of the fact that BPS solutions transform under generalized spectral ows into other BPS solutions [6], and of the fact that BPS solutions develop an AdS3 UV region when certain of their moduli are put to zero, in order to perform a systematic search for solutions with a NHEK infrared and an AdS3 UV. At rst glance, both these procedures should be automatically successful. The asymptotics of the solutions is controlled by 17 parameters, and if one sets to zero the constant, 1 and cos2 terms in g and certain divergent components of the elds one should obtain a 10-parameter family of solutions that have the leading radial component of the metric of the form d 22 , which is the hallmark of an asymptotically-AdS solution. However, things are not so simple. Most of the solutions obtained in this class have closed timelike curves, and if one tries to naively impose all the conditions that eliminate the closed timelike curves none of the asymptotically-AdS solution seem to survive. Similarly, if one tries to obtain these solutions by relating BPS and almost-BPS solutions (as we will explain in detail in appendix A) at the end one has to solve 7 equations for 17 variables, which however appear at rst glance to have no solution. Thus, despite the presence of a large number of available constants, neither of the two hunting methods we use seems to be very willing to yield asymptotically-AdS solutions. Fortunately, a careful analysis reveals that things are not so bleak. Indeed, we nd that among the many ways to solve the constraints associated to the absence of closed timelike curves, there is one that produces nontrivial solutions that are asymptotically AdS3 S3=ZN . Furthermore, the second method produces exactly the same solution, which we take as a remarkable con rmation that we have really identi ed the way to construct solutions that have an AdS3 factor in the UV and NHEK in the IR. The existence of such a class of ows has several important implications. 1Other approaches to understanding the microstates of Kerr black holes include e.g. [9]. 2See [10{12] for approaches along similar lines. { 2 { The rst is for the debate whether the theory dual to the NHEK geometry can be described as the infrared limit of a local CFT2 or only of a non-local one. Since the NHEK geometry can be obtained by a certain identi cation of an (uncompacti ed) warped AdS3 geometry, it was argued in [13] that the theory dual to NHEK is the DLCQ of a nonlocal theory which is dual to warped AdS3 (oftentimes known as a \dipole" quantum eld theory). Another proposal is that the dual to NHEK is given by the identi cation of a more exotic type of conformal eld theory, called warped-CFT (wCFT) [14{16], which would be a local theory [17]. Our construction shows that the theory dual to the NHEK geometry could be equally well obtained as the IR xed point of many RG ows of \vanilla CFT's" and hence it can be understood without resorting to wCFT's or dipole needs to turn on to ow to a NHEK infrared,3 and in particular whether this deformation is similar to the one that triggers the RG ow of the asymptotically AdS3 the near-horizon AdS3 of a BPS black ring [20], or whether it is rather a deformation of the Lagrangian. Thus, the existence of this family of ows opens a new route for determining S3 solution to what the CFT dual to 5D NHEK is. In addition to embedding the NHEK-containing solutions of [ 1 ] in AdS3, we also construct the full form of their R-R three-form eld strength. These solutions can be obtained by dualizing the twice-spectrally- owed almost BPS solutions that were constructed in the M2-M2-M2 duality frame in [21] to the D1-D5-P duality frame. In [ 1 ] this duality transformation was performed for the metric and the dilaton, which was enough to ascertain the existence of these solutions and to perform some basic regularity checks. However, to perform all the regularity checks and to be able to understand all the properties of these solutions one must also construct this three-form explicitly. As we will see in appendix B, even if this construction involves several duality transformations that act rather nontrivially on the R-R elds, the nal implicit form of the expression that give threeform eld strength is quite simple. However, its explicit form is much more complicated than for BPS and almost-BPS solutions, even after making several simplifying assumptions (equation (2.21)). Besides its importance for the programme of embedding the NHEK geometry in String Theory, the calculation of the R-R three-form also lls an important gap in our knowledge of almost-BPS solutions and generalized spectral ows thereof. Indeed, the full solution that comes from applying three generalized spectral ows on an almost-BPS solution has so far only been constructed in the M2-M2-M2 duality frame [21]. Writing some of these solutions in the D1-D5-P duality frame allows one to embed these solutions into six-dimensional 3And conversely what is the deformation of the NHEK geometry that allows it to UV- ow to a nice local CFT in the UV [13]. { 3 { ungauged supergravity and explore whether these solutions belong to a larger class of wiggly solutions, as it happens when supersymmetry is preserved [22, 23]. This paper is organized as follows. In section 2 we review the family of supergravity solutions that contain NHEK regions in the infrared and that can be obtained by a sequence of generalized spectral ow transformations from almost-BPS solutions with D1, D5, momentum and KK monopole charges. We then nd the explicit R-R elds for these solutions whose derivation is given in appendix B. In section 3 and appendix A we develop two di erent systematic procedures to search for solutions with an AdS UV, and identify a sub-class of the large family of supergravity backgrounds with a NHEK infrared constructed in [ 1 ] that have an AdS3 S3=ZN asymptotic region. 2 In nite families of NHEK embeddings in String Theory In [ 1 ] it was shown that an in nite family of IIB supergravity solutions with a NHEK infrared can be obtained by performing generalized spectral ow transformations [6, 21] on a class of non-supersymmetric, \almost-BPS", multicenter supergravity solutions [7, 8] whose charges correspond to D1 and D5 branes, momentum and KK monopoles. However, in [ 1 ] only the metric and dilaton have been constructed. While this was enough to ascertain the existence of such solutions, in order to perform all regularity checks, calculate their asymptotic charges or use holography to read o the features of the UV CFT, the explicit expressions for the R-R elds are needed. It is the purpose of this section to complete the construction of the supergravity solutions that contain a NHEK infrared by explicitly computing these R-R elds. 2.1 Almost-BPS D1-D5-P-KK solutions The ve-dimensional Hodge star is with respect to the time- bration over the four-dimensional base space ds42 = V 1(d + A)2 + V d 2 + 2(d 2 + sin2 d 2) with ?3 dA = dV; (2.3) 4We work in units where 0 = gs = 1 and use the conventions of [21]. This solution can be derived from a sequence of duality transformation acting on an M2-M2-M2 solution where the three M2 branes wrap that the relative sign in the metric (2.1) between dz and A3 di ers from the one in [ 25 ]. We explain the reason for this in appendix B where we explicitly carry out the sequence of duality transformations on the spectrally owed M2-M2-M2 solution to get the spectrally owed D1-D5-P solution reviewed in section 2.2. { 4 { The metric of the extremal BPS and almost-BPS D1-D5-P-KK solutions is [ 20, 24, 25 ]4 ds2 = 1 Z3pZ1Z2 (dt + k)2 + pZ1Z2ds42 + p (A3 in (2.3) speci es the orientation of the Taub-NUT base and distinguishes between BPS and almost-BPS solutions [7]. We will consider almost-BPS solutions corresponding to the minus sign in (2.3) for which A = qKK cos d . The one-forms AI consist of an \electric" and a \magnetic" part: AI = dt + k ZI + aI ; with the warp factors ZI with I = 1; 2; 3 encoding respectively the asymptotic electric D1, D5 and momentum charges and the vector potentials aI encoding the local magnetic dipole charges. In the base space (2.3) the magnetic one-form potentials aI and the angular momentum one-form k can be decomposed as (2.4) (2.5) (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) HJEP05(216) aI = KI (d + A) + wI ; k = (d + A) + ! ; where KI and are functions and wI and ! are one-forms in the three-dimensional at space. Solution speci ed by ZI , aI , k are obtained by solving the almost-BPS equations 2 d ?3 dZI = CIJK V d ?3 d(KJ KK ) ; ?3dwI = KI dV V dKI ; ?3d! = d( V ) V ZI dKI ; d ?3 d(V ) = d(V ZI ) ?3 dKI : where CIJK = j IJK j. Acting with d?3 on (2.9) yields an equation for : and obtained by solving5 The solutions ZI and denote by LI and M . to (2.7) and (2.10) contain harmonic functions which we will An almost-BPS solution is then determined by the functions (V; fKI g; fLI g; M ). In anticipation of section 2.2 we also de ne [21] the one-forms vI ?3dvI = dZI 2 CIJK [V d(KJ KK ) KJ KK dV ] ; ?3d = ZI dKI KI dZI + V d(K1K2K3) K1K2K3dV: 2.2 Non-BPS D1-D5-P-KK solutions with a NHEK infrared Applying a sequence of supergravity transformations known as generalized spectral ows [6] to the solution (2.1){(2.2) yields a large class of solutions that contains the NHEK geometry as an infrared limit. We refer to [21] for details about the generalized spectral ow 5Note that we renamed the one-form v0 of [21] to since we already use v0 as the constant in the Taub-NUT potential. { 5 { transformations and summarize here the solution for the NS-NS elds [ 1 ]. We then go on to compute the R-R elds of the spectrally owed solution. The spectrally owed extremal D1-D5-P-KK metric is [21]: ds2 = e 1 q Ze3 dt + ek + eaI ; decomposed as and the magnetic one-form potentials eaI and the angular momentum one-form ek can be e aI = PeI (d + Ae) + weI ; ek = e(d + Ae) + !e ; with !e = !. The one-forms weI and Ae in the three-dimensional at space are given by Ae = A I wI CIJK 2 J K vI + 3 ; e wI = wI + CIJK J vK CIJK 2 J K ; (2.17) and functions ZeI , WfI , PeI , e and Ve are given by (2.13) (2.14) (2.15) (2.16) (2.18) ZeI = WfI = PeI = e = Ve = NI V e 1 1 NI V 2 e ; 2 NI CIJK 2 CIJK 2 T 3V + CIJK J K TI ZI 2 CIJK I J TK ZK V ZI TI KI + I ZJ ZK (2T1 1)V CIJK J2 K2 TI2ZI2 2 CIJK I J K TJ ZJ TK ZK T 3V (CIJK J K TI ZI ) + T 6V 2 + 8 3T 3V ; CIJK 2 + e ; WfI 1=2 ; 3 Z3 + J K ZI TI V I V TJ ZJ TK ZK + T 3V 2 ; where T = (T1T2T3)1=3 and = ( 1 2 3)1=3 and we de ned TI = 1 + I KI ; NI = 2 CIJK I2ZJ ZK + V TI2ZI 2 I V TI : (2.19) In order to complete the solution of [ 1 ] we need to construct the R-R three-form ux elds and are summarized appendix B. The implicit expression for this ux has a similar form as that of almost-BPS solutions: where the ve-dimensional Hodge star is with respect to the time- bration over the fourdimensional base space (2.14). HJEP05(216) 2.3 Towards the full explicit form of the R-R elds Despite its apparent simplicity, the explicit expression of Fe(3) in terms of the harmonic functions determining the solutions is a very complicated nested expression. In particular the action of the ve-dimensional Hodge star involves repeated use of several of the tilded forms and functions and the application of the almost-BPS equations (2.7){(2.12). The purpose of this subsection is to give the complete explicit form of this eld for a certain sub-class of solutions. We will restrict ourselves to solutions with = 0 (yet keeping ! 6= 0), that only have constant terms in the KI harmonic functions (corresponding to Wilson lines along the Taub-NUT direction in ve dimensions and to axion vev's in four dimensions) but no poles. The class of solutions whose explicit three-form eld we nd does not include the asymptotically-AdS3 solutions that are the main focus of this paper. However, we hope that the (rather complicated) expression we nd will be an important stepping stone for nding the three-form of that more complicated class of solutions. To facilitate the calculation we note that while the rst two generalized spectral ows with parameters 1 ; 2 act non-trivially on the solution the third spectral ow with parameter 3 corresponds to a coordinate transformation. Hence, without loss of generality we can set 3 = 0 (which implies that T3 = 1). To obtain an explicit expression for the three-form ux from (2.20) we express all ve-dimensional Hodge stars in terms of threedimensional ones and make successive use of the almost-BPS equations (2.7){(2.12). We refer to appendix B for the details. The explicit expression for the three-form ux is: We now explore the asymptotics of the supergravity solutions of section 2 and, in particular, whether it is possible to construct a geometry with an AdS3 ultraviolet and a NHEK infrared. We will pursue two strategies. The rst is to investigate various limits of the parameters controlling the UV of these solutions in order to produce an AdS factor in the metric. The asymptotics of the solutions is controlled by 17 parameters. Setting to zero the constant, 1 , cos and cos2 terms in g and certain divergent components of the elds should yield a 10-parameter family of solutions whose leading radial component of the metric is of the form d 2 2 characteristic of AdS solutions. The second strategy, which we will present in appendix A, makes clever use of the relations between the asymptotics of BPS and almost-BPS solutions and turns out to give exactly the same class of solutions as the rst strategy. 3.1 Solutions with NHEK infrared The supergravity solutions of section 2.2 can have multiple centers where the geometry near one of the centers is NHEK. Our starting point to look for solutions with AdS asymptotics is a two-center solution where in addition to the non-BPS D1-D5-P-KK black hole that becomes the NHEK geometry in the infrared we add another smooth center corresponding to a supertube. The asymptotics of this solution are prototypical for all solutions in the class constructed in [ 1 ] and, as we will see, the conditions that ensure that the nal solution is asymptotically-AdS3 do not depend on the particular distribution of centers and charges in the infrared. One can ask whether there is any reason behind our strategy to try to embed the NHEK solution in an asymptotically-AdS3 solution that has two or more centers, and hence topological-nontrivial three-cycles. Our original inspiration came from three-charge BPS black ring solutions [24, 26, 27] embedded in an asymptotically AdS3 S3 solution [20]: these rings have another AdS3 region in the vicinity of the black ring center, with smaller AdS radius; thus the black ring solution can be thought of as an RG ow from a CFT in the UV to a CFT with lower central charge in the IR. The full solution has a topologicallynontrivial three-sphere at whose North Pole this smaller AdS3 sits. One can also see by direct calculation that a multicenter solution is necessary to get a NHEK region in the infrared. One could try to start from a single-center solution and { 8 { play with the constants in the harmonic functions to obtain a cohomogeneity-one solution with an AdS3 UV, but one will nd that the infrared of this solution is always AdS3. This is essentially because in a single-center solution the KI harmonic functions are constant,6 and hence their value is the same at in nity and at the black hole. The only way to get something other than AdS3 in the infrared is to makes those values di erent by introducing additional sources for KI , and thus additional centers. Hence, we begin by considering a two-center solution with a Taub-NUT base containing a non-BPS black hole and a supertube at distance R in the R3 base of the Taub-NUT space. The solution is easiest to describe using two sets of spherical coordinates, one centered at the black hole ( ; ) and another centered at the supertube ( ; ), related by = p 2 + R2 2 R cos ; cos = cos R ; cos 0 = R cos ux of the solution are given by (2.13) and (2.20) of section 2.2 and are speci ed by the functions V; KI ; LI ; M; and the one-forms wI ; !; vI ; .7 The functions V; KI ; LI ; M are qKK dI ; ; M = m0 + LI = `I0 + J + QI + j eI + c : cos 2 ; The supertube charges, dipole charges and angular momentum are denoted by feI g = fe1; e2; 0g, fdI g = f0; 0; d3g and j = e21de32 , and the black hole has charges fQI g = fQ1; Q2; Q3g and angular momentum J . The KK gauge potential is A = qKK cos d . The warp factors are given by ZI = LI . The angular momentum one-form k = (d + A) + ! is given by8 = ! = M V + `03d3 + 2V Q3d3v0 + + J cos + j cos 2 2 `03d3v0 cos (qKK`30 + v0Q3)d3 cos 0 d : Q3d3qKK cos V c R sin2 6In an almost-BPS solutions the KI functions cannot have poles where the Taub-NUT harmonic function V has poles, as these give rise to singularities or diverging moduli [28]. 7In section 6 of [8] the solution for a non-BPS black ring and a black hole in Taub-NUT was found in the type IIA duality frame. We adapt this solution for our purpose, by turning o a charge and two dipole charges of the black ring to transform it into a supertube. 8Note that in the expression for given in [8] there is a typo: the factor of cos in the last term in (3.3) is missing. { 9 { qKK R qKK R ; Regularity requires the absence of Dirac strings at the poles = 0; of the two-sphere which implies that ! must vanish there: 0 = ! =0 = 0 = ! = = + J + s j s 2 qKK R 2R + Q3d3v0 2R : between the centers in the R in the ve-dimensional solution. The second equation is interpreted as the bubble equation that determines the distance R 3 base of the Taub-NUT space, or the radius of the supertube 3.2 Hunting for AdS asymptotics We now explore the UV structure of the metric (2.13) speci ed by the functions in section 3.1. To simplify the analysis we make use of the fact that the third spectral ow corresponds to a coordinate transformation and so we can set 3 = 0 without loss of generality. On the other hand, for the IR to correspond to NHEK, 1 and 2 must be non-vanishing. For the UV to be AdS we need the radial part of the metric to behave as g d 2= 2 at large . From (2.13) and (2.14) we read o the radial part of the metric: and the one-forms ; vI in (2.17) are and v3 = (Q3 + k10k20qKK) cos d : (3.6) (3.7) where we made use of (2.18). The large- expansion of the NI gives q g = Ve Ze1Ze2 = pN1N2 ; NI = nI + I = + I = 2 + O(1= 3) ; (3.8) (3.9) (3.10) (3.11) (3.12) (3.13) with similarly for 2), and imposing In general this expression contains terms proportional to cos that would give rise to warped AdS asymptotics, and since our purpose is to obtain \normal" AdS we require these terms to vanish. To leading order this can be achieved by writing I = I0 + I cos , 1 corresponding respectively to the rst and second line of (3.16) (and The leading radial part of the metric is then simply given by and similar expressions for n2; 2 ; 2 (obtained by exchanging 1 $ 2), where tI = 1 + kI0 I is de ned by expanding the function TI = tI + O(1= ). To have the constant and 1= terms in (3.12) vanish in the large expansion requires After imposing these constraints the leading-order term in the radial part of the metric becomes g g p (3.14) (3.15) (3.16) cos ; (3.17) (3.18) (3.19) (3.20) (3.21) (3.22) (3.23) (3.24) subject to relations between the parameters of the solution following from the constraints (3.17) and (3.19). Regularity imposes further constraints: for absence of closed timelike curves (CTCs) we need as well as Z1Z2Z3V !2 > 0 ; 2V 2 + (CTC)0 + (CTC)1= + (CTC)2= 2 + > 0 ; where in the second line we have expanded the no-CTC condition in the UV. Since the constants `I0 are non-negative and the charges QI , e1, e2 in ZI are strictly positive, to avoid CTCs we also need the constant v0 to be non-negative and the KK monopole charge qKK to be strictly positive. Altogether we have: `I0; v0 > 0 ; QI ; e1; e2; qKK > 0 : The condition (3.22) is equivalent to the existence of a global time function [29]. In the following we will solve the constraints (3.17) and (3.19) of section 3.2. It is useful to consider separately the solutions where `03 = 0 and `03 6= 0. 3.3.1 One can then solve n1 = 0 to obtain The second constraint n2 = 0 is solved by negative leading-order contribution (CTC)0 = have to be imposed: [n1 = n2 = (CTC)0 = 0] , Further solving 1 = 0 for d3 gives In principle we have to consider both solutions. However, it turns out that canceling the m20 + `01`02`03v0 in (3.23) both conditions v0 = `30 t11t22 ; `02 = With this, the constraint 2 = 0 is automatically satis ed. For the metric to correspond to unwarped AdS at large we have to impose (3.19) which can be achieved by setting c = ( R 2 + ` 0 For the solution to be free of CTCs we need to impose (3.22) subject to the constraints (3.28), (3.29) and (3.30). The leading non-vanishing term is (CTC)2= 2 which, evaluated at = 0 or = , is: 0 6 (CTC)2 = 1 4 (`30)2 t2 2 (Q1 + e1) + (`10)2 t1 1 qKK + 2 t2 t1 Q3 1 (Q2 + e2) 2 2 + 2(`10)2 t 2 1 12 qK2K + 2 (3.25) (3.26) (3.27) (3.28) (3.29) however, kills the sought after d 2 = 2 term in the metric. Hence, the solutions in section 3.1 with `03 6= 0 do not have an asymptotic AdS metric. When `03 vanishes, the non-negativity of the leading part of the no-CTC condition (3.22) imposes: , 2 = 0 one can see that one must also have `01 = ` The constraints n1 = n2 = 0 are then automatically satis ed. To satisfy the no-CTC condition (3.22) at order 1= 2 implies (CTC)2 c + Q3d3(qKK=R + v0=2) 2 sin4 > 0 , c = d3Q3(Rv0 + 2qKK) : 2R Finally, the constraints 1 = 2 = 0 are satis ed by imposing9 tI = 0 , kI0 = 1= I for I = 1; 2 : Notice that this implies that T1 = T2 = 0.10 With these constraints the bubble equation (3.11) becomes (3.33) (3.34) (3.35) (3.36) (3.37) where we used the usual relation between the supertube charges, dipole charges and angular momentum. The distance between the centers, R, is determined once the constants controlling the UV are xed. With these constraints the leading radial metric component is d 2 = 2 and the ultraviolet is free of CTCs. To ensure that there are no CTCs in the infrared we expand (3.22) for small : 0 6 Q1Q2Q3qKK c2 cos2 ; which can always be satis ed by taking Q1; Q2 su ciently large. The no-CTC conditions (3.21) are automatically satis ed near the black hole. It is also trivial to check that the determinant of the UV metric at leading order is constant and negative: Det(gUV) 12(Q2 + e2) 22(Q1 + e1) 3 Q32 sin2 < 0 : 9One can also satisfy 1 = 2 = 0 by setting v0 = 0. This would, however, set j = 0 which is inconsistent since the supertube has non-vanishing angular momentum. 10This is a consequence of having only one non-vanishing dipole charge (d3 in K3). Replacing the supertube by a black ring turns on extra dipole charges, d1 and d2, in respectively K1 and K2 and in such a solution T1 6= 0, T2 6= 0. 3.4 We now summarize the features of the solutions that contain a NHEK region [ 1 ] and have an AdS3 UV. The complete bulk metric is given by (2.13) of section 2.2 and is speci ed by the harmonic functions (3.2) of section 3.1. For the UV of the metric (2.13) to have a leading radial component d 2 = 2 we have to constrain the constants that appear in the harmonic functions that determine the solution: Absence of CTCs requires positivity of the charges (3.24) and c = This equation enforces the cancelation of the sin2 term in the asymptotics of !, and is necessary for avoiding CTCs only when the asymptotics is AdS. This cancelation is equivalent to the vanishing of the total four-dimensional angular momentum (JR) of the solution. In the particular solution we consider, this angular momentum has both a contribution from the black hole as well as a contribution coming from the interaction between the magnetic dipole charge of the supertube and the electric charge of the black hole, and equation (3.39) forces these contributions to cancel each other. For a more general multicenter solution containing one or more black holes (whose near-horizon regions are transformed by the generalized spectral ows into a NHEK region), there will be more contributions to JR (see for example [30]), and we expect that the only requirements to have asymptotically-AdS3 solutions are that the constants entering in the harmonic functions satisfy (3.38) and that the total JR vanishes. Of course, the more general multicenter solutions will also have to satisfy the corresponding bubble equations, which for the two-center solution we focus on are quite simple: but in general will be much more complicated [8]. In terms of a more standard radial coordinate r2 = 4qKK the UV metric at leading order is ds2UV where (3.38) (3.39) + 2 1 lUV r 2 4lU2V ( dT 2 +dY 2)+lU2V (d 2 +sin2 d 2)+ d N ( dT + dY )2 N (dT + 3dY ) sgn( 1 1) cos d d N sgn( 1 1) cos d cos ( dT 2 + dY 2) + UV ds2T 4 ; T = s Q3v0 1 2 t ; Y = s Q3v0 1 2 v0 t + z k 0 3 + d3 R N sgn( 1 2) ; (3.42) N = j 1 2jQ3 ; lU2V = N p(Q1 + e1)(Q2 + e2) ; = d3pQ3qKKv0 ; = qKK e1 Q1 +e1 + e2 Q2 +e2 2 R ; UV = The term in (3.41) proportional to corresponds to a spectral ow and so we can remove it by a coordinate transformation: = N 2 2lU4V (T + 3Y ) ; and we de ned where (3.43) 1 2 s Q2 +e2 Q1 +e1 : (3.44) 2 (3.45) (3.46) lUV ; S3=ZN lUV T 4 with subleading perturThe leading part of the UV metric (3.45) is AdS3 bations that trigger the RG ow to NHEK. This establishes that all the geometries containing NHEK regions in the infrared [ 1 ] can be embedded into asymptotically-AdS3 solutions, and hence that the theories dual to NHEK can arise as xed points of Renormalization Group ow from the D1-D5 CFT dual to AdS3. Di erent multicenter solutions will correspond to di erent RG ows, and we leave the detailed study of these ows for future work. r 2 4lU2V ( dT 2 +dY 2)+lU2V (d 2 +sin2 d 2)+ d N + 0( dT + dY )2 2 0dY (dT + dY ) 0 cos ( dT 2 + dY 2) + UV ds2T 4 ; sgn( 1 2) cos d Acknowledgments We would like to thank Monica Guica for valuable discussions and collaboration in the early state of this project and Geo rey Compere and Diego Hofman for interesting discussions and comments on the manuscript. The work of I.B. was supported in part by the ERC Starting Grant 240210 String-QCD-BH, by the John Templeton Foundation Grant 48222 and by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-RFP31321 (this grant was administered by Theiss Research). The work of A.P. is supported by National Science Foundation Grant No. PHY12-05500. I.B. and A.P. are grateful to the Centro de Ciencias de Benasque Pedro Pascual and to the Aspen Center for Physics for hospitality and support via the National Science Foundation Grant No. PHYS-1066293. Throats with a NHEK | an alternative approach In section 3 we carried out a systematic analysis for embedding the multicenter solutions containing NHEK regions in an asymptotic AdS geometry and identi ed a set of constraints that have to be satis ed. We now describe a second approach that yields the same set of constraints. The leading and subleading asymptotics of an almost-BPS solution is controlled by 17 parameters, and the same is true for its BPS equivalent. Generalized spectral ows shu e these parameters in a certain way, and at the end one has to x 7 of these parameters in order to obtain an AdS3 UV. Since the near-horizon of any almost-BPS black hole can be transformed into a NHEK geometry by choosing the appropriate generalized spectral ow parameters [ 1 ], this implies that there should be at least a 10-parameter family of solutions with NHEK IR and an AdS3 UV. However, as emphasized in the Introduction, most of the solutions have closed timelike curves, which have to be eliminated (as we also did in the approach described in section 3). At the end we will nd that this approach only produces one way to get a solution that is asymptotically AdS3 and free of CTCs and this is exactly the same solution as the one obtained in section 3! The method. The procedure which we will now develop makes clever use of the fact that the only di erence between the asymptotics of BPS and almost-BPS solutions is the sign of one of the electric elds [7], of the fact that BPS solutions transform under generalized spectral ows into other BPS solutions [6], and of the fact that BPS solutions develop an AdS3 UV region when certain of their moduli are put to zero.11 The procedure we follow is summarized in this illustration: BPS: AdS3 UV # 2 BPS gSFs BPS+ non-BPS: NHEK IR and AdS3 UV? UV " 2 non-BPS gSFs almost-BPS We start with a BPS solution that has an asymptotic AdS3 geometry. We then perform two BPS generalized spectral ow transformations (gSF) that yields another BPS solution whose asymptotics we identify with those of the almost-BPS solution of section 3.1 (to distinguish the BPS and almost-BPS solutions whose asymptotics we are matching we use, respectively, + and sub/super scripts). This xes some of the moduli of the almost-BPS solution. Applying two non-BPS generalized spectral ow transformations to this almostBPS solution yields a non-BPS solution which contains a NHEK geometry in the IR and whose UV we show to correspond to an AdS3 geometry. P Hunting for AdS3. In a BPS solution that is determined by 8 harmonic functions [27, 31, 32] one can read o the asymptotics and charges from H = h + P i ii , where h is the vector of constants and tot = lim !1 i ii is the total charge vector as determined by 11As in section 3, without loss of generality, we set to zero the third spectral ow. the harmonic functions H = (V; fKI g; fLI g; M ): tot = vtot; fkItotg; f`tIotg; mtot : A BPS solution has AdS3 asymptotics if the vector of constants takes the form [33]: with l03 6= 0. The requirement that12 h tot; hAdSi = 0 xes m0 = kt3ot=vtot with vtot 6= 0. Performing two BPS spectral ow transformations with parameters 1 and 2, corresponds to the supergravity transformations [6]:13 M ! M+ = M ; LI ! LI+ = LI KI ! KI+ = KI 2 I M ; CIJK J LK + CIJK J K M ; V ! V+ = V + I KI 1 1 2 CIJK I J LK + 3 CIJK I J K M : This yields another BPS solution with the new asymptotics: h2gSF AdS = For (A.4) to correspond to the asymptotics of the almost-BPS solution of section 3.1 we have to identify up to order 1= : V+ V ; Z+ I ZI ; (aI+ k+) (aI k ) ; k+ k : (A.5) The gauge potentials, warp factors and angular momentum of the almost-BPS solution are given in section 3.1. Those of the BPS solution are given by a I+ = K+ V+ I (d ZI+ = LI+ + + = M+ + + A+) + wI+ ; CIJK KJ+KK+ ; 2 V+ 2 V+ 1 KI+LI+ + CIJK KI+KJ+KK+ ; 6 V+2 with the constants in the harmonic functions H+ = fV+; fKI+g; fLI+g; M+g speci ed by (A.4). The identi cations (A.5) then become [34] ; K+ I V+ + Z+ I K I Z I : (A.9) 12The symplectic product is given by h tot; hi = 2(vtotm0 mtotv0) + (kItot`0 I `totkI0). I 13Note that work in the conventions of [21] which have an extra factor of 2 multiplying M+ when compared to the conventions of [6]. (A.1) (A.2) (A.3) (A.6) (A.7) (A.8) This yields the following relations between the BPS and almost-BPS solutions (up to order 1= ): from which we can read o h+ = (v0;+; fkI0;+g; f`I0;+g; m0;+): V+ K+ I L+ I M+ CIJK KI+KJ+KK+ ; ; + v02; m0; CIJK kJ0; `0J; kK0; `0; K 4v0; m0; `10; `02; `0; 3 Note, that as in section 3.3 we can use the bubble equation and the supertube relation between charges, dipole charge and angular momentum to obtain the relation: From h+ = h2gSF AdS with (A.4) we can solve for the constants of the almost-BPS solution: v0; = ` 0; I = 0 ; = = I CIJK (`0J; kJ0: v0; 2m0; )(`0K; kK0: v0; 2m0; ) ; v0; `0J; `0; K 8m03; We can use this to solve for l03 in (A.19) and plug the result into the expression for k30; . Since m0 is a free parameter, k30; is in fact unconstrained. With this remark and dropping the \ " in the constraints (A.19) we nd that the constants entering in the harmonic functions determining the solution are: (A.10) (A.11) (A.12) (A.13) (A.14) (A.15) (A.16) (A.17) ! : (A.18) (A.19) (A.20) which agrees precisely with the results (3.38) of our analysis in section 3.3. The extra condition on c in (3.39) follows from the requirement that there be no closed time-like curves | a condition we also have to impose here. Hence, the method described here yields exactly the same solution as the one discussed in section 3.3 and we take this as a remarkable con rmation that we have really identi ed the ow from AdS3 to NHEK. B In [ 1 ] it was shown that a large class of supergravity solutions with a NHEK infrared can be obtained by dualizing the twice-spectrally- owed almost-BPS solutions that were constructed in the M2-M2-M2 duality frame in [21] to the D1-D5-P duality frame. In [ 1 ] this duality transformation has been performed on the metric and dilaton, but not the R-R elds. We now complete this class of supergravity solutions by computing the R-R elds in the D1-D5-P duality frame. From M2-M2-M2 to D1-D5-P. The metric and three-form gauge potential of the twice-spectrally owed M2-M2-M2 solution obtained in [21] are given by where dTI denote the volume forms of the three two-tori TI2 and ds21 = dx24 + dx25, ds22 = dx26 + dx27, ds23 = dx28 + dx29, denote the metrics on the latter. We use the shortcut notation (Z~1Z~2Z~3)1=3 and all tilded quantities are de ned in (2.17){(2.18). To bring this solution to the D1-D5-P duality frame we have to perform a Kaluza-Klein reduction on one of the torus legs followed by a sequence of three T-duality transformations. Note that this parallels the chain of dualities derived in [ 25 ] for dualizing BPS solutions. • KK reduction on x9. The Kaluza-Klein reduction of the solution (B.1) along the x9 direction (renaming the remaining leg of the torus x10 z) yields14 (B.1) (B.2) F (4) = dC(3) + dB(2) ^ C(1) = dA~1 ^ dT1 + dA~2 ^ dT2 : 14Note that the sign in the expression for B(2) in (B.2) depends on which torus leg we compactify; if we chose to reduce along x10 rather than x9 the minus sign in (B.2) would turn into a plus sign. Anticipating that B(2) will become part of the metric and three-form ux after the nal T-duality along z (B.7) this explains the relative sign between dz and, respectively, A3 in (2.1) and Ae3 in (2.13). C(1) = 0 ; 1 Z~3pZ~1Z~2 1 4 = log Z~1Z~2 ~ Z3 ; (dt + k~)2 + qZ~1Z~2ds42 + s ~ Z1 Z~2 ds12 + s ~ Z2 Z~1 ds22 + pZ~1Z~2 dz2; ~ Z3 B(2) = ~ A3 ^ dz ; C(3) = A~1 ^ dT1 + A~2 ^ dT2 ; ? (dA 10 e2 ^ dx 5 ^ dx 7 ^ dx ) = 8 ? dA 5 e2 ^ ^ dz dx ; 6 (B.4) Z ~5 2 Z Z ~2 ~3 where the ve-dimensional Hodge star is with respect to the ve-dimensional metric • T-duality along x ds 2 10 F log Z Z ~ ~ ~ 1 2 3 Z Z ~3 1 Z Z ~ ~2 • T-duality along z ds 2 10 F (3) = = = 1 2 ~ Z 3 log 1 p Z ~5 2 Z Z ~2 ~3 3 1 ~ ~ Z Z 1 2 ~ Z ~ Z 1 2 ; 1=4 q (dt + k) + ~ 2 ~ ~ Z Z B F (2) (4) = = ~ A 3 ^ dz ; Z ~5 2 Z Z ~2 ~3 3 1 p ~ ~ Z Z Z 3 1=4 s ~ Z ~ Z 1 2 (dt + k) + ~ 2 ~ ~ 2 Z Z ds + 1 2 4 p ~ Z 3 ~ ~ Z Z Z ~ Z 1 2 (dz A ) + 3 ~ 2 2 2 (ds + ds ) ; 1 2 B (2) = 0 ; ? dA 5 e2 dA e1 ^ (dz ~ A ) : 3 • T-duality along x C F log ~ ~ ~ Z Z Z Note that the ten-dimensional Hodge star can be expressed as (dt+k) + Finally the R-R three-form ux in the D1-D5-P duality frame is given by F e (3) = 5 Z e2 Note that this expression has the same form as the R-R three-form ux (2.2) of the almostBPS solution. However, (B.8) is considerably more complicated. In particular, the action (B.5) (B.7) (B.8) of the ve-dimensional Hodge star on this expression involves repeated use of expressions of the tilded forms and functions, as well the simpli cation of the result using the almost-BPS equations (2.7){(2.12). To give the explicit form we make some simplifying assumptions. We will restrict ourselves to solutions with = 0 (yet keeping ! 6= 0), and allow the KI to be constants (dKI = 0). Computing the explicit form of the three-form ux for more general solutions is much more complicated and we will not address it here. As mentioned before, the third spectral ow corresponds to a coordinate transformation in the D1-D5-P duality frame, so without loss of generality we only focus on the solution obtained after two spectral ows and set 3 = 0. This leads to considerable simpli cations. Since T3 = 1 + 3K3 and K3 = const < 1 we have T3 = 1 and the tilded functions (2.17){(2.18) relevant here simplify to Ve = jT1T2V 1 2Z3j ; Wf1 = N1 T1T2V 1 2Z3 ; e = Wf2 = and we write where Pe1 = N1 G1 + e ; Wf1 ( 1T2Z2 + 2T1Z1)Z3V ; T1T2V Pe2 = N2 V 2 e 1 2Z3 N2 G2 + e ; Wf2 ; Wf3 = N3 T1T2V + 1 2Z3 Pe3 = N3 G3 + e ; Wf3 G1 = T1K1Z1V + 1Z2Z3 ; G2 = T2K2Z2V + 2Z1Z3 ; G3 = K3Z3V: (B.12) We recall the form of the gauge potentials (2.15) (B.9) ; (B.10) (B.11) (B.13) (B.14) (B.15) (B.16) dek ! Wf2 ; AeI = weI + PeI (d + Ae) To give the explicit form of Fe(3) we trade all ve-dimensional Hodge stars for threedimensional Hodge stars and subsequently use the almost-BPS equations (2.7){(2.12) to replace as many of the three-dimensional Hodge stars as possible. First term: ?5 dAe2. We want to express 1=4 Ze25 e 1 q Ze3 ?5 dAe2 = ?5 dw2 + dPe2 ^ (d + Ae) + Pe2dAe + d ^ (dt + ek) whose ve-dimensional Hodge stars are with respect to the metric q ds52 = in term of three-dimensional Hodge stars with respect to the at three-dimensional base spanned by yi with i = 1; 2; 3. We introduce the vielbeine e e0 = f0(dt + ek) ; e e1 = f1(d eei = fidyi ; where With and thus 1 W f2 ? d 3 W f2 1=4 f = (Z Z ) 0 e1 e2 1=4 Z e3 1=2 ; f = (Z Z ) 1 e1 e2 +1=4 V 1=2 ; f = (Z Z ) i e1 e2 +1=4 +1=2 V e : (B.17) ? (e 5 ? (e 5 ? (e 5 i i 1 e e ^ ^ ^ e e ^ 1 1 e ^ i e ) = f f 0 i ijk ^ ijk ijk j 2 e j e e ^ k e e ^ k e ; ^ 0 e ; k e ; e ? (dy 5 i ^ j 1 dy ) = f f f (dt + k) (d + A) 0 1 i e ^ e ^ ijk k dy ; ? dy 5 i ^ 1 (dt + k) = f f f (d + A) 0 1 i e ^ ? (d + A) dy 5 e ^ i ^ ijk 2 dy j (dt + k) ; ^ e k dy ; 1 = f f f (dt + dk) 0 1 i e ^ 1 f f f (dt + dk) (d +A) e 0 1 i e ^ e With this, the Hodge star combinations appearing in (B.14) are ? dA = 3 e T T dV 1 2 3 f2 ? dW = (V ? dZ 3 3 the terms in (B.14) can be expressed as d ^ e 1 (dt+k) = f f f (d + A) 0 1 i e ^ 1 1 1 ? dw = f f f 5 2 0 1 i e ^ (dt + k) (d + A) ? dw ; 3 2 e e ? dP 5 e2 ^ (d + A) = f f ? dW 3 f2 W f2 2 ; ? d 3 W f2 e V Z 3 T T V 1 2 Finally we get for (B.14): ? dA = (dt + !) (d + A) ^ e ^ (V ? dZ 3 3 Z ? dV ) + 3 3 ( T ? dZ + 1 2 3 2 dz). The second term in equation (B.8) is given by dAe1 ^ (Ae3 dz) = d (dt + ek) + Pe1(d + Ae) + we1 ^ (dt + ek) + Pe3(d + Ae) + we3 dz : (B.25) Using (B.9){(B.12) we can write (B.25) as dAe1 ^ (Ae3 dz) = (dt + !) ^ (we3 dz) ^ d + (d + Ae) ^ (we3 dz) ^ d (dt + ek) + 1 Wf3 G1 N1 N3 G3 + e Wf3 G3 N3 d 1 Wf1 (d +Ae) ^ where, using the almost-BPS equations (2.7){(2.12) one can write dAe = T1T2 ?3 dV 1 2 ?3 dZ3 ; d!1 = K1T2 ?3 dV + 2 ?3 dZ3 : (B.27) Putting (B.24) and (B.26) together we arrive at the following expression for the R-R eld strength: + (w3 dz) e (B.28) (B.29) e where e and we recall from section 2.2: As a check, a tedious but straightforward exercise shows that the expression (B.28) is closed. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. JHEP 03 (2013) 028 [arXiv:1203.4227] [INSPIRE]. Phys. 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