Regular 3-charge 4D black holes and their microscopic description
IIPhT-T14/066
Prepared for submission to JHEP
Regular 3-charge 4D black holes and theirmicroscopic description
Iosif Bena and C. S. Shahbazi
Institut de Physique Théorique, CEA Saclay.
E-mail: [email protected] , [email protected] Abstract:
The perturbative α (cid:48) corrections to Type-IIA String Theory compactified on a Calabi-Yauthree-fold allow the construction of regular three-charge supersymmetric black holes in fourdimensions, whose entropy scales with the charges as S ∼ (cid:0) p p p (cid:1) . We construct anM-theory uplift of these “quantum” black holes and show that they can be interpreted asarising from three stacks of M2 branes on a conical singularity. This in turns allow us relatethem via a series of dualities to a system of D3 branes carrying momentum and thus togive a microscopic interpretation of their entropy. a r X i v : . [ h e p - t h ] J un ontents N = 2 four-dimensional supergravity and the H-FGK formalism 13B A quantum class of black holes 16 String theory has proven to be extremely successful in reproducing the entropy of super-symmetric black holes, such as the three-charge black hole in five dimensions [1] and thefour-charge black hole in four dimensions [2–4]. The entropies of these black holes scale likethe square root of the product of their charges (or some some duality-invariant form thereof[5]), and in the microscopic counting this square root comes from using Cardy’s formula tocount the states of a certain 1+1 dimensional system of strings and branes.However, in certain four-dimensional compactifications of string theory one can con-struct three-charge black holes whose entropy scales with these charges like S ∼ (cid:0) p p p (cid:1) [6]. The curvature at the horizon of these black holes is small, precisely as one wouldexpect from the fact that their entropy grows like the square of the charge. These blackholes cannot be constructed in “normal” four-dimensional supergravity, where the horizoncurvature of three-charge black hole is Planckian, but exist if one adds to the supergravityLagrangian certain terms coming from perturbative String-Theory α (cid:48) corrections at treelevel in g S . Since these black holes do not have a regular limit when these correction termsare removed they are called “quantum black holes.” Unlike other black hole solutions constructed in these α (cid:48) -corrected theories[7–10] – 1 –he purpose of this letter is to try to understand the microscopic entropy of the typeIIA quantum black holes constructed in [6]. The first step in this direction is to upliftthese black hole solutions to eleven dimensions and to propose an M-theory interpretationin terms of three mutually-orthogonal intersecting stacks of M2-branes at the tip of a four-dimensional Gibbons-Hawking-like base. Since the α (cid:48) corrections to Type-IIA Calabi-Yaucompactifications come from higher-derivative terms in String Theory [11, 12], the upliftedblack hole will be a solution of the equations of motion of eleven-dimensional supergravitymodified by the addition of certain higher-derivative terms.The second step is to argue that if the Calabi-Yau manifold can be written as a (possiblysingular) elliptic fibration, the branes that make the eleven-dimensional solution can bedualized to a configuration of two intersecting mutually-supersymmetric D3 branes carryingmomentum along their common direction. By counting the possible way of carrying thismomentum and by remembering that these D3 branes sit on top of a conical singularity thateffectively enhances the central charge of the 1+1 dimensional theory on their worldvolume,we are able to reproduce the peculiar charge dependence, S ∼ (cid:0) p p p (cid:1) , of the entropy ofthe quantum black holes.In section 2 we introduce the effective theory corresponding to Type-IIA String Theorycompactified on a Calabi-Yau manifold as well as the corresponding Type-IIA quantumblack hole solutions. In section 3 we argue that the M-theory uplift of quantum black holescan be interpreted as arising from three stake of intersecting M2 branes and in section 4 weuse this to propose a microscopic description of their entropy. Section 5 contains conclusionsand future directions. In the appendices (A) and (B) we review the construction of Type-IIA quantum black holes as well as the H-FGK formalism [13–21] and the structure of N = 2 four-dimensional ungauged supergravity coupled to vector multiplets that were usedin their construction. Quantum black holes [6] are solutions of the effective theory corresponding to Type-IIAString Theory compactified on a Calabi-Yau manifold in the presence of perturbative cor-rections to the Special Kähler geometry of the vector multiplet sector. Despite having onlythree charges, these four-dimensional supersymmetric black holes have a macroscopicallylarge horizon area. It is not hard to see either from this or from their explicit constructionthat these black holes do not have a macroscopic horizon in the “classical” limit, when theperturbative corrections are turned off, which justifies calling them “quantum black holes”.
Type-IIA String Theory compactified to four-dimensions on a Calabi-Yau three-fold,with Hodge numbers ( h , , h , ) , is described, up to two derivatives, by a N = 2 four-dimensional supergravity coupled to vector- and hyper-multiplets. As explained in appendix(A), we are going to truncate the hyperscalars to a constant value, and therefore we willonly be concerned about the vector-multiplet sector of the theory. The corresponding– 2 –repotential can be written as an infinite series around (cid:61) m z i → ∞ , and it is given by[22–25] F = − κ ijk z i z j z k + ic i (2 π ) (cid:88) { d i } n { d i } Li (cid:16) e πid i z i (cid:17) , (2.1)where z i , i = 1 , ..., n v = h , , are the scalars in the vector multiplets, κ ijk are the classicalintersection numbers, d i ∈ Z + is a h , -dimensional summation index and Li ( x ) is thethird polylogarithmic function. The constant c is proportional to the Euler characteristicof the Calabi-Yau three-fold, χ multiplied by the Riemann zeta function: c ≡ χζ (3)(2 π ) .The first two terms in the prepotential correspond to tree level and fourth-loop pertur-bative (which is the only non-vanishing one) contributions in the α (cid:48) -expansion, respectively[25] F P = − κ ijk z i z j z k + ic , (2.2)while the third term comes from non-perturbative corrections produced by world-sheetinstantons. In this paper we will focus on large-volume compactifications where thesecorrections can be safely ignored, and hence focus on the quantum black holes obtainedfrom the prepotential (2.2). The most general quantum black hole solutions, that aregoverned by the prepotential (2.1) have been constructed in [21].Our starting point is the prepotential (2.2), which in homogeneous coordinates X Λ , Λ =(0 , i ) , can be written as F ( X ) = − κ ijk X i X j X k X + ic (cid:0) X (cid:1) . (2.3)The scalars z i are given by z i = X i X . (2.4)Adding the constant term c to the prepotential modifies the geometry of the scalar manifold,which is no longer homogeneous, and therefore it is said that the geometry has been corrected by quantum effects. The scalar geometry defined by (2.3) is hence the so-called quantumcorrected d -SK geometry [26, 27]. The attractor points of (2.3) have been extensively studiedin [28].The Type-IIA quantum black holes belong to a particular class of purely magneticblack hole solutions of the theory defined by (2.3). We review them in appendix (2) andrefer the reader to [6, 21] for more details. Type-IIA quantum black holes exist in any Type-IIA Calabi-Yau compactification with h , > h , . To make contact to a description of these black holes in terms of intersecting Actually, the prepotential obtained in a Type-IIA Calabi-Yau compactification is symplectically equiv-alent to the prepotential (2.1). – 3 –ranes we have to choose a particular model and the best candidate is the popular STUmodel, whose black hole solutions and attractors have been extensively studied. Since thehypermultiplets are truncated, the specific value of h , is irrelevant as long as it is smallerthan h , . The values of h , and κ ijk are h , = 1 , κ = 1 (2.5)and therefore the prepotential (2.3) becomes F ( X ) = X X X X + ic (cid:0) X (cid:1) . (2.6)The “normal” four- and five-dimensional black holes and attractors of this quantum-corrected STU model been previously considered in [7–10, 19, 28], but here we focus on theblack holes that do not have a classical ( c → ) limit. The solution corresponding to theseblack holes (discussed in detail in Appendix B) has e − U = 3 c (cid:12)(cid:12) H H H (cid:12)(cid:12) / , (2.7) z i = ic H i ( H H H ) / . (2.8)The space-time metric is therefore ds = − − c − (cid:12)(cid:12) H H H (cid:12)(cid:12) − / dt + 3 c (cid:12)(cid:12) H H H (cid:12)(cid:12) / d(cid:126)y , (2.9)where d(cid:126)y = (cid:0) dy (cid:1) + (cid:0) dy (cid:1) + (cid:0) dy (cid:1) is the Euclidean metric on R . Since in the H-FGKformalism the H -variables correspond to the imaginary part of the covariantly holomorphicsymplectic section appropriately weighted to be Kähler neutral, supersymmetry requirethem to be harmonic functions on the transverse space R . A single-center black hole hasthen H i = a i + p i √ r , i = 1 , , , (2.10)where r = (cid:0) y (cid:1) + (cid:0) y (cid:1) + (cid:0) y (cid:1) , the p i are the three charges of the black hole (A.12) andthe a i are arbitrary constants that can be written in terms of the asymptotic value of thescalars at spatial infinity z i ∞ as a i = − s p i Im z i ∞ √ c , (2.11)where s p i is the sign of the charge p i . The entropy of this black hole is S = 3 c π (cid:12)(cid:12) p p p (cid:12)(cid:12) / (2.12)and its mass is the sum of the three charges:– 4 – = (cid:114) c (cid:0) a a p + a a p + a a p (cid:1) . (2.13)It is easy to see that each term that contributes to the mass is positive definite since Sign a i = Sign p i and a a a > . In the next section we will try to describe this three-charge four-dimensional black hole in terms of intersecting branes in M-theory. In order to see whether Type-IIA quantum black holes have an interpretation via intersect-ing branes it is desirable to have the precise ten-dimensional configuration correspondingto the four-dimensional solution. For tree-level Type-IIA Calabi-Yau compactifications themap between the ten-dimensional and the four-dimensional fields is known [29–32], but sincewe are considering four-dimensional solutions to the prepotential that includes perturbativecorrections in α (cid:48) at tree level in g S , and since these correction come from an R -like term inten dimensions, no explicit map is known. However, as we will show below, we will still beable to control the behavior of the dilaton and the Calabi-Yau volume, which will allow usto obtain the higher-dimensional configuration corresponding to Type-IIA quantum blackholes. As explained in appendix (A), for black hole solutions of ungauged four-dimensional su-pergravity, the hyperscalars are truncated to a constant value. In principle, the dilatonbelongs to the universal hypermultiplet, and therefore one may naively conclude that itshould be constant for every black hole solution. However, in the process of obtaining theeffective N = 2 four-dimensional supergravity in its standard form (A.1), several rescal-ings and redefinitons are performed on the original ten-dimensional fields. In addition,since we are considering all the perturbative corrections to the Special Kähler sector, thecorresponding N = 2 ungauged supergravity action is not the effective compactificationtheory of Type-IIA String Theory at tree level, so we should expect more intrincate redef-initions. Our purpose is to show now that Type-IIA quantum black holes have a constantten-dimensional dilaton and a constant Calabi-Yau manifold volume, and we will do this intwo steps: Tree level:
At tree level, the Type-IIA dilaton φ is related to the four-dimensional dilaton q as follows [33, 34] e − q = e − φ (cid:112) Vol , (3.1)where Vol = 16 (cid:90) J ∧ J ∧ J , (3.2)– 5 –s the volume of the Calabi-Yau manifold, J being the corresponding Kähler form. TheKähler potential of the N = 2 Special Kähler manifold is related to the volume of thecompactification Calabi-Yau manifold by [33, 34] e −K = e φ (cid:90) J ∧ J ∧ J = e φ Vol . (3.3)Since for Type-IIA quantum black holes both e −K and q are constants, this implies thatboth e − φ (cid:112) Vol = const . and e φ Vol = const . (3.4)and hence at tree level both the dilaton φ and the Calabi-Yau volume are constant. Perturbative corrections:
It is also easy to see that this tree level result is not changedwhen including perturbative corrections. Indeed, [11, 12] have shown that the loop correc-tions in ten dimensions that give rise to the perturbative corrections of the prepotential fromthe four-dimensional point of view, only mix the dilaton and the volume among themselves.Therefore, since they were constant at the tree level, they continue to be constant afterthe loop corrections have been taken into account. We thus conclude that for Type-IIAquantum black holes the dilaton and the volume of the Calabi-Yau manifold are constant.
In order to argue that the M-theory uplift of quantum black holes can be interpreted ascoming from a superposition of M2 branes on a conical singularity it is useful to recall theusual supersymmetric solution corresponding to three stacks of M2 branes on a six-torus[35, 36] ds = ( H H H ) (cid:104) − ( H H H ) − dt + H − (cid:0) dx + dx (cid:1) + H − (cid:0) dx + dx (cid:1) + H − (cid:0) dx + dx (cid:1) + g mn dy m dy n (cid:3) , (3.5)where m, n = 1 , . . . , and the M2-branes are respectively located along the (cid:8) x , x (cid:9) , (cid:8) x , x (cid:9) and (cid:8) x , x (cid:9) directions. Supersymmetry requires the transverse metric, g mn , to beHyper-Kähler and the M2 warp factors to be harmonic in this metric: ∆ y H i ( y ) = 0 , i =1 , , . To compare with the uplifted quantum black holes we will focus on solutions wherethe Hyper-Kähler space is a Gibbons-Hawking (Taub-NUT) space: ds = V ( y ) − (cid:0) d Ψ + A i ( y ) dy i (cid:1) + V ( y ) δ ij dy i dy j , i = 1 , , , (3.6)where ∗ dA ( y ) = dV ( y ) ⇒ ∆ y V ( y ) = 0 . (3.7)The eleven-dimensional metric is therefore given by– 6 – s = ( H H H ) (cid:104) − ( H H H ) − dt + H − (cid:0) dx + dx (cid:1) + H − (cid:0) dx + dx (cid:1) + H − (cid:0) dx + dx (cid:1) + ds (cid:3) , (3.8)where now the H i are harmonic in R . In order to interpret Type-IIA quantum black-holes as composed of three stacks of or-thogonally intersecting M2-branes, one should uplift their solution to M-theory. Since thefour-dimensional theory where these black holes are constructed includes all the perturba-tive corrections in α (cid:48) , at tree level in g S , the uplifted black holes should be solutions ofthe standard eleven-dimensional supergravity to which one has added the higher-derivativeterms that give rise to the 4D perturbative α (cid:48) -corrections.There are two features of the quantum black hole metric (2.9) that will guide us toobtain this solution. The first is that the volume of the six-dimensional torus/Calabi-Yau manifold is constant to all levels in the corrections and hence, rescaling the time to t → (cid:112) c t , the eleven-dimensional solution can be put into an M2-brane form: ds = ( H H H ) (cid:20) − (cid:16) c H H H (cid:17) − dt + H − (cid:0) dx + dx (cid:1) + H − (cid:0) dx + dx (cid:1) + H − (cid:0) dx + dx (cid:1) + d ˜ s (cid:3) (3.9)The four-dimensional base metric d ˜ s is no longer Gibbons-Hawking but becomes: d ˜ s = 3 − c − (cid:0) H H H (cid:1) − d Ψ + 3 c (cid:0) H H H (cid:1) δ ij dy i dy j , i = 1 , , . (3.10)This metric is not Ricci flat and it does not even have constant curvature. This is tobe expected, since the solution (3.9) solves the equations of motion of eleven-dimensionalsupergravity modified by appropriate higher curvature terms, which modify in turn theGibbons-Hawking character of the base. Indeed, if one tries to compare this metric to aGibbons-Hawking one, one finds that on one hand the gauge field A i ( y ) , corresponding toD6 brane charge in ten dimensions, is zero, but that on the other hand the correspondingwarp factor is not constant by rather has the form: V ( y ) = 3 c (cid:0) H H H (cid:1) , (3.11)which has the same behavior at infinity and near the black hole as the warp factor of aTaub-NUT space with a nontrivial charge: lim r → V ( y ) ∼ r , (3.12)where r = (cid:113) ( y ) + ( y ) + ( y ) . Furthermore, when the three warp factors become equalthe function V becomes harmonic throughout the space– 7 – y V ( y ) = 0 , (3.13)despite the absence of a Gibbons-Hawking (D6) charge.It important to notice that in the M-theory uplift the term ds is constant, which is con-sistent with the ten-dimensional dilaton of the quantum black hole solution being constant.This is an nontrivial check that the eleven-dimensional brane configuration we propose givesthe fundamental constituents of Type-IIA quantum black holes. Having obtained an eleven-dimensional metric that resembles that of three stacks of coin-cident M2 branes, we can easily compactify it along one of the torus directions to obtain aD2-D2-F1 metric, which upon a further T-duality along the F1 direction becomes a D3-D3-P metric, where the momentum P runs along the direction common to the two D3 branes.This duality chain transforms the quantum eleven-dimensional black hole into a type IIBD3-D3-P black hole, whose microscopic entropy can be reproduced straightforwardly byStrominger-Vafa-type arguments. Indeed, if the numbers of the two types of D3 branes are N and N and if N P units of momentum are running along the common directions of thesebranes, the most efficient way to carry this momentum when N and N are co-prime isto use the strings stretched between the two stacks of D3 branes. These “bi-fundamental”strings have a mass gap equal to N N R , where R is the radius of the common directionof the branes, and hence the entropy of the system comes from partitioning the N P unitsof momentum between modes that carry integer multiples of N N , or otherwise from thenumber of integer partitions of N N N P . By taking into account the fact that there arefour bosonic species of bi-fundamentals as well as their fermionic partners, this gives theentropy π √ N N N P .The argument above reproduces the entropy of a D3-D3-P black hole in five dimensions,which is sourced by a stack of branes in R . One can write this R as a Gibbons-Hawkingspace with V = 1 /r and a nontrivial fiber satisfying dA = (cid:63)dV . If we focus instead on astack of D3-D3-P branes in a Taub-NUT space with Kaluza-Klein monopole charge N , thecorresponding warp factor is V = 1 + Nr and we obtain a regular four-dimensional blackhole with entropy π √ N N N P N [2, 3].We would like to use a similar argument to explain microscopically the entropy ofour quantum black holes. However, at first glance there are two problems with this: Thefirst is that our base, (3.11), is no longer Gibbons-Hawking but has A = 0 and V =3 c ( H H H ) / . Nevertheless, near the branes the warp factor behaves like that of aGibbons-Hawking space. Hence, even if the branes do not sit on top of an A N singularity,they sit on some other conical singularity whose effect on the central charge of the D3-D3CFT one can calculate.The second problem is that a generic eleven-dimensional uplift of a quantum black holewill not be a six-torus but a more complicated CY manifold. The key ingredient neededto relate the quantum black holes to the D3-D3-P system is the presence of two U (1) – 8 –sometries, one of which is used for reducing to a ten-dimensional Type IIA black hole, andthe other for T-dualizing . This can be easily done for any CY manifold that has a T fiber. Nevertheless, in order for our construction of quantum black holes to yield regularsolution, this elliptic fibration must be singular: CY manifolds with regular fibration havezero Euler characteristic and hence c vanishes which makes the black hole horizon singular.This singularity can be cured by including non-perturbative α (cid:48) effects, again at tree-levelin g S [21], but the resulting solutions involve the Lambert W function and are much harderto manage.The way out is to focus on singular elliptic fibrations, which give CY manifolds withnonzero Euler number. The places where the fibration degenerates become seven-branesupon dualization to the type IIB duality frame. The presence of these seven-branes doesnot affect the entropy counting, because this entropy comes from D3-D3 strings carryingmomentum, which do not see the seven-branes.There are two ways to take into account the effect of the conical singularity on theentropy. The first is to compare this singularity with a Gibbons-Hawking solution, anddetermine its effective Gibbons-Hawking charge. The second is to focus on the near-horizongeometry of the black hole and to compute the corresponding Brown-Henneaux centralcharge [37], which determines how the central charge of the D3-D3 CFT increases when theD3 branes are placed on top of the conical singularity. As we will explain below, the twocalculations are equivalent, but since the first is more intuitive we will present it here.Near the tip of a Gibbons-Hawking metric ds = V ( y ) − (cid:0) d Ψ + A i ( y ) dy i (cid:1) + V ( y ) (cid:16) dr + d Ω (cid:17) , i = 1 , , , (4.1)with V = 1 + Nr (4.2)the metric becomes that of R / Z N ds ∼ dρ + ρ d ˜Ω , (4.3)with ρ = 2 √ r and d ˜Ω the standard metric on S / Z N . When the D3-D3 system is placedat the tip of this space its central charge increases by a factor of N . This is a well-knownphenomenon for the D1-D5-P black hole in Taub-NUT [38], and our system is just its T-dual. Now, given a conical metric of the type (4.3), there is a way to extract directly thisfactor: N = V S V S / Z N , (4.4)where V S is the volume of the three-sphere S and V S / Z N is the volume of the S / Z N ,at the same radius. In the Brown-Henneaux formalism this ratio of the volumes also gives If these two isometries are not present our microscopic description does not work, but neither does themicroscopic description of “normal” M2-M2-M2 black holes. – 9 –he decrease of the effective three-dimensional Newton’s constant, and hence the increaseof the central charge of the corresponding CFT.For the quantum black hole metric we discussed in section 3.3, the base metric d ˜ s = V ( r ) − d Ψ + V ( r ) (cid:16) dr + r d Ω (cid:17) , V ( y ) = 3 c (cid:0) H H H (cid:1) , i = 1 , , (4.5)has a conical singularity in the near-tip region: d ˜ s ∼ √ r cp p p ) / d Ψ + 3 (cid:0) cp p p (cid:1) / √ r (cid:16) dr + r d Ω (cid:17) . (4.6)Upon defining (with hindsight) N E ≡ √ (cid:0) √ c p p p (cid:1) / √ , (4.7)and changing coordinates to ( ρ = 2 √ N E r ), the metric near the conical singularity becomes d ˜ s ∼ dρ + ρ (cid:32) d Ψ N E + d Ω (cid:33) ≡ dρ + ρ d Ω transverse . (4.8)Hence, the singularity will increase the central charge of the D3-D3-P CFT by a factorgiven by the ratio of the volume of S and the volume of the transverse space, which isnothing but N E . V S V Ω transverse = 2 π π N E π = N E . (4.9)Hence, the microscopic entropy of the quantum black holes will be given by S = 2 π (cid:112) N N N P N E , (4.10)where N and N are the numbers of D3 branes and N P is the number of momentumquanta. Since the supergravity charges are proportional to these numbers, it is clear thatthis microscopic entropy count reproduces the correct charge growth of the quantum blackhole S = 32 c π (cid:12)(cid:12) p p p (cid:12)(cid:12) / , (4.11)which is already an important confirmation that our strategy is correct. Of course, the idealwould be to find exactly the coefficients that relate the supergravity charges to the quantizedones, and therefore establish that the macroscopic and the microscopic entropies are iden-tical. However, since we know neither the eleven-dimensional uplifts of the Maxwell fieldsof the four-dimensional quantum black hole, nor the volume of the Calabi-Yau three-foldand its submanifolds, we cannot determine these coefficients from first principles. However,– 10 –hat we can do is to use the symmetry of the STU model in order to argue that in the M2-M2-M2 duality frame the supergravity charges are related to the quantized brane numbersvia the same proportionality constant, γ : ˜ p = γN , ˜ p = γN , ˜ p = γN P , (4.12)where we have defined ˜ p i ≡ p i √ to ease the presentation. We can now ask what is the valueof γ that makes the microscopic and the macroscopic entropies agree. Upon using (4.12),equation (4.11) becomes: S = 2 π (cid:115) c ˜ p ˜ p ˜ p c (˜ p ˜ p ˜ p ) = 2 π (cid:115) c γ N N N P N E (4.13)and therefore the desired value of γ is: γ ≡ c − . (4.14)There is a nontrivial check that this value satisfies: The number N E , which gives theincrease of the CFT central charge is expected to be a natural number, at least for somevalues of N , N and N P . However, N E is defined in terms of c , and therefore containsboth a factor of ζ (3) as well as square and cubic roots: N E = √ (cid:0) √ c ˜ p ˜ p ˜ p (cid:1) / . (4.15)Hence one may naively infer that N E can never be a natural number. However, it turnsout that when expressing N E in terms of quantized charges using (4.14) both the ζ (3) andthe √ drop out: N E = (4 N N N P ) / (4.16)and therefore N E can easily be an integer for a suitable choice of N , N and N P . Thefact that the transcendental number in c drops out of this relation is a nontrivial checkof our proposed microscopic description. Indeed, in [7] it was argued that a black holeentropy expression that contains such a transcendental number can never be reproducedfrom microscopic calculations, and our microscopic proposal evades this by absorbing thistranscendental number into the central charge increase of the underlying CFT.The puzzling aspect of equation (4.16) is that it makes the “classical” limit very hardto see. Indeed, as we explained in section (2) if one turns off the quantum corrections( c → ) while keeping the supergravity charges of the black hole constant, the horizonbecomes singular. This is reflected in our construction by the fact that our black hole hasthree charges and, when c → , the warping of the base space becomes trivial and thecorresponding entropy becomes that of the three-charge system in four dimensions, whichdoes not give rise to a macroscopic horizon.However, one can also ask what happens if one takes the c → limit while keepingthe quantized black hole charges, N , N and N P constant. This does not affect N E , so it– 11 –ooks like in this limit the black hole entropy remains macroscopic. This is consistent withthe fact that in this limit the four-dimensional supergravity charges blow up (4.14), but thefactors of c cancel from the expression of the near-horizon limit of the metric (2.9), whichremains a regular AdS × S . On the other hand, the expressions of the four-dimensionalmoduli diverge, and therefore it appears that in this limit the dictionary between ten- andfour-dimensional solutions breaks down. It would clearly be interesting to try to deriveequation (4.14) from first principles to see precisely how this breakdown occurs. We have constructed an eleven-dimensional metric (3.9) that upon dimensional reductiongives the Type-IIA quantum black hole of the STU quantum-corrected model. Because thefour-dimensional metric has constant dilaton and CY volume, this eleven-dimensional upliftcan be interpretation as arising from three stacks of orthogonal M2 branes that sit at theapex of a cone in a four-dimensional transverse space. Because of the presence of correctionterms in the Lagrangian, this space is not Gibbons-Hawking, although it has exactly thesame kind of warping as a Gibbons-Hawking space. The strength of the conical singularityis proportional to the cubic root of the product of the three M2 chargesWhen the dix-dimensional internal space of the compactification has a T fiber we candualize this solution to an asymptotically four-dimensional three-charge D3-D3-P solutionin type IIB string theory. In flat space the microscopic entropy of this system is not enoughto give rise to a regular horizon, but we have shown that the conical singularity enhancesthe central charge of this system, and the resulting microscopic entropy reproduces theentropy of the of the quantum black hole: S = c π (cid:12)(cid:12) p p p (cid:12)(cid:12) / up to an overall coefficientwhich we could not determine. We have however been able to show that if this coefficientis such that the entropies match, a certain dependence of the entropy on transcendentalnumbers drops out, which we believe is a nontrivial check of our proposal.Since we are working in a supergravity theory in the presence of quantum corrections tothe geometry of the scalar manifold, our proposed microscopic description is not at the samelevel of rigor as the usual three- and four-charge black hole entropy counting. However, thefact that we have found a brane interpretation that reproduces the highly-unusual chargedependence of the entropy of quantum black holes makes us confident that we have identifiedthe correct microscopic framework for understanding the entropy of all Type-IIA quantumblack holes, which remains as an important open problem in String Theory. Acknowledgments
We would like to thank T. Ortín for useful discussions and comments. The work of IB wassupported in part by the ERC Starting Independent Researcher Grant 240210, String-QCD-BH, by the John Templeton Foundation Grant 48222: “String Theory and the AnthropicUniverse” and by a grant from the Foundational Questions Institute (FQXi) Fund, a donoradvised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-– 12 –FP3-1321 to the Foundational Questions Institute.The work of CS was supported by theERC Starting Independent Researcher Grant 259133, ObservableString. A N = 2 four-dimensional supergravity and the H-FGK formalism Type-IIA quantum black holes are black hole solutions of Type-IIA String Theory com-pactified down to four dimensions on a Calabi-Yau three-fold, which is described, up totwo derivatives, by a N = 2 four-dimensional ungauged supergravity. Therefore, it is con-venient to review the basic formulation of the theory and its vector multiplet sector, sincethe hypermultiplets and the fermions can be always truncated for black hole solutions. Thebosonic sector of any N = 2 four-dimensional supergravity coupled to vector multiplets canbe written as follows [39, 40] S = (cid:90) d x (cid:112) | g | (cid:110) R + G i ¯ j ( z, ¯ z ) ∂ µ z i ∂ µ ¯ z ¯ j + 2 I ΛΣ ( z, ¯ z ) F Λ µν F Σ µν − R ΛΣ ( z, ¯ z ) F Λ µν (cid:63) F Σ µν (cid:111) (A.1)The z i ( i = 1 , ...n v ) denote the n v complex scalar fields of the vector multiplets, whichparametrize an n v -dimensional Special Kähler manifold with Kähler metric G i ¯ j ( z, ¯ z ) . F Λ = dA Λ denote the field strengths of the Λ = 0 , ..., n v one-form connections A i that belongto the vector multiplets, plus the graviphoton A . The real matrices I ΛΣ ≡ Im N ΛΣ ( z, ¯ z ) , R ΛΣ ≡ Re N ΛΣ ( z, ¯ z ) denote respectively the imaginary, negative definite, and real partsof the symplectic complex period matrix N . Hence, the period matrix determines thecouplings of the one-form connections A Λ to the scalars z i of the vector multiplets. Theequations of motion following from the action (A.1) are given by G µν + 2 G i ¯ j [ ∂ µ z i ∂ ν ¯ z ¯ j − g µν ∂ ρ z i ∂ ρ ¯ z ¯ j ] + 8Im N ΛΣ F Λ + µρ F Σ − νρ = 0 , (A.2) ∇ µ ( G i ¯ j ∂ µ ¯ z ¯ j ) − ∂ i G j ¯ k ∂ ρ z j ∂ ρ ¯ z ¯ k + 12 ∂ i [ G Λ µν ∗ F Λ µν ] = 0 , (A.3) ∇ ν ∗ G Λ νµ = 0 , (A.4)where we have defined G Λ ≡ − √− g δSδ ∗ F Λ = Re N ΛΣ F Σ + Im N ΛΣ ∗ F Σ . (A.5)The Maxwell equations for the field strengths F Λ together with the corresponding Bianchiidentities can be written in terms of differential forms as follows dG Λ = 0 , dF Λ = 0 . (A.6)Notice that, since G Λ is a closed two-form, it can be written locally as the exterior derivativeof a one-form A Λ – 13 – Λ = dA Λ . (A.7)where A Λ is the so-called magnetic dual of A Λ and both sets of connection one-forms canbe arranged into a symplectic vector A M = (cid:0) A Λ , A Λ (cid:1) T . Supersymmetry constrains thecouplings of all the fields of the theory in a very precise way which, for the vector-multipletsector, is elegantly encoded in the language of Special Kähler Geometry [39, 41]. In fact, thebosonic Lagrangian of N = 2 four-dimensional supergravity coupled to vector multiplets isdetermined by choosing a holomorphic section Ω ∈ Γ ( SV ) or, equivalently (when it happensto exist), a homogeneous function F ( X ) of degree two, the N = 2 prepotential , from which G i ¯ j and N ΛΣ can be easily obtained as G i ¯ j = − ∂ i ∂ ¯ j log (cid:8) i (cid:2) ¯ X Λ ∂ Λ F − X Λ ∂ Λ ¯ F (cid:3)(cid:9) , (A.8) N ΛΣ = ∂ ΛΣ ¯ F + 2 i Im( ∂ ΛΛ (cid:48) F ) X Λ (cid:48) Im( ∂ ΣΣ (cid:48) F ) X Σ (cid:48) X Ω Im( ∂ ΩΩ (cid:48) F ) X Ω (cid:48) . (A.9)Here X Λ denote the homogeneous coordinates on the scalar manifold, related to the scalarfields z i via z i ≡ X i X , (A.10)Therefore, choosing a second-degree homogeneous function F ( X ) automatically determinesan N = 2 , four-dimensional, ungauged supergravity theory coupled to vector multiplets,which has the appropriate matter content for constructing black hole solutions. The mostgeneral static and spherically symmetric metric that solves the equations of motion (A.1)is given by [42–44] ds = − e U ( τ ) dt + e − U ( τ ) γ mn dx m dx n ,γ mn dx m dx n = r sinh r τ (cid:104) r sinh r τ dτ + dθ + sin θdφ (cid:105) , (A.11)where τ is the radial coordinate. When (A.11) describes a physical black hole solution, r isthe non-extremality parameter , the exterior of the event horizon corresponds to τ ∈ ( −∞ , ;the event horizon is at τ = −∞ and spatial infinity corresponds to τ → − . The innerpart of the Cauchy horizon corresponds to τ ∈ ( τ s , ∞ ) , with the inner horizon at τ → ∞ and the singularity at τ = τ s for a certain positive and finite real number τ s [16]. Sincethe metric is spherically symmetric, we will assume that all the fields of the theory dependexclusively on the radial coordinate τ . We define the black hole charges as p Λ = (cid:90) S ∞ i ∗ F Λ , q Λ = (cid:90) S ∞ i ∗ G Λ , (A.12)where S ∞ denotes an space-like two-sphere at spatial infinity τ → , p Λ correspond to themagentic charges and q Λ correspond to the electric charges of the black hole, which can betogether arranged into a symplectic vector Q M ≡ (cid:0) p Λ , q Λ (cid:1) T . In the background given by– 14 –A.11) Maxwell’s equations can be integrated explicitly, in such a way that the completeelectric connection one-form A Λ is given in terms of the time component A Λ t of the electricconnection one-form and the time component of the magnetic connection one-form A Λ t .Indeed, let Σ M ≡ ( A Λ t , A Λ t ) T be a symplectic vector made from the time components ofthe electric A Λ and magnetic A Λ connection one-forms. Then, it can be shown that Σ M = 12 (cid:90) e U M MN Q N dτ , (A.13)where M MN is a symplectic and symmetric matrix constructed from the couplings of thescalars and the vector fields as M MN ( N ) ≡ (cid:0) I + RI − R (cid:1) ΛΣ − (cid:0) RI − (cid:1) Λ Σ − (cid:0) I − R (cid:1) Λ Σ (cid:0) I − (cid:1) ΛΣ . (A.14)We choose to express all Maxwell field strengths in terms of the time components of theelectric and the magnetic connection one-forms. For the electric field strengths this gives: F Λ tτ = − ∂ τ A Λ t , F Λ θφ = sin θ e − U (cid:18)(cid:0) I − (cid:1) ΛΣ dA Σ t dτ − (cid:0) I − R (cid:1) Λ Σ dA Σ t dτ (cid:19) , (A.15)and the expression for the magnetic field strengths G Λ can be similarly obtained fromequation (A.15) using equation (A.5). Since the connection one-forms can be explicitlyintegrated, they can be eliminated from the action. The four-dimensional N = 2 ungaugedsupergravity action coupled to vector multiplets can then be shown to be completely equiv-alent, assuming the space-time background given (A.11) and radial dependence for all thefields, to the one-dimensional effective FGK action [43] for the n v complex fields z i ( τ ) andthe real field U ( τ ) S FGK [ U, z ] = (cid:90) dτ (cid:110) ˙ U + G i ¯ j ˙ z i ˙¯ z ¯ j − e U V bh ( z, ¯ z, Q ) (cid:111) , (A.16)together with the Hamiltonian constraint , ˙ U + G i ¯ j ˙ z i ˙¯ z ¯ j + e U V bh ( z, ¯ z, Q ) = r . (A.17)Here V bh is the so-called black hole potential , which is given by [43] V bh ( z, ¯ z, Q ) ≡ M MN ( N ) Q M Q N . (A.18)We are now ready to introduce the H-FGK formalism. The H-FGK formalism [14, 16, 17,20, 44] consists of a particular change of variables from the (2 n v + 1) -real (cid:0) U, z i (cid:1) to a newset of (2 n v +2) -real variables H M ( τ ) which transform as a symplectic, linear, representationthe U-duality group of the theory, and become harmonic functions in Euclidean R in thesupersymmetric solution. The equations of motion in the new variables H M ( τ ) can bewritten as – 15 – ∂ P MN log W (cid:20) ˙ H M ˙ H N − Q M Q N (cid:21) + ∂ P M log W ¨ H M − ddτ (cid:18) ∂ Λ ∂ ˙ H P (cid:19) + ∂ Λ ∂H P = 0 (A.19)together with the Hamiltonian constraint − ∂ MN log W (cid:18) ˙ H M ˙ H N − Q M Q N (cid:19) + (cid:32) ˙ H M H M W (cid:33) − (cid:18) Q M H M W (cid:19) − r = 0 (A.20)where Λ ≡ (cid:32) ˙ H M H M W (cid:33) + (cid:18) Q M H M W (cid:19) , (A.21)and e − U = W ( H ) ≡ ˜ H M ( H ) H M , ˜ H M + iH M = V M X , (A.22)with V M being the covariantly holomorphic symplectic section that determines the vector-multiplet sector of N = 2 supergravity, and X a complex variable with the same Käh-ler weight as V M , making the quotient V M /X Kähler invariant. The symplectic vector ˜ H M ( I ) ≡ ˜ H M ( H ) stands for the real part of V M written as a function of the imaginarypart, H M ; this can always be done by solving the stabilization equations . The function W ( H ) is usually known in the literature as the Hesse potential .The effective theory is now expressed in terms of n v + 1) variables H M . The solutiondepends on n v + 1) + 1 parameters, namely the n v + 1) charges Q M = (cid:0) p Λ , q Λ (cid:1) T and the non-extremality parameter r , from which it is always possible to reconstructthe complete solution in terms of the four-dimensional fields of the theory. The H-FGKformalism introduces an extra real degree of freedom. Hence the H-FGK action enjoysan extra gauge symmetry which, by gauge fixing, allows to get rid of the extra degree offreedom [17]. B A quantum class of black holes
In this appendix we present the solution of the equations (A.19) and (A.20) that correspondto the quantum black holes of Type-IIA String Theory. Type-IIA quantum black holes arebased on the following truncation of the H -variables and the charges H = H = H i = 0 , p = q = q i = 0 . (B.1)Using now equation (B.1) together with equations (A.22) and (2.3) we find e − U = W ( H ) = (3! c ) / (cid:12)(cid:12)(cid:12) κ ijk H i H j H k (cid:12)(cid:12)(cid:12) / , (B.2)where c = χζ (3)(2 π ) . From equation (B.2) it follows that, in order to have a non-singularmetric, c must be positive, that is, h , > h , is a necessary condition in order to obtain an– 16 –dmissible solution. There are plenty of Calabi-Yau manifolds that satisfy this condition,so we will not worry any more about it. The scalar fields, purely imaginary, are z i = i (3!) c H i (cid:16) κ ijk H i H j H k (cid:17) / , (B.3)It is easy to see that the solution is not consistent in the classical limit c → , and alsothat no classical limit can be assigned to it, since when c = 0 the model is already singularbefore solving the equations of motion. Hence, we conclude that the corresponding solu-tions are genuinely quantum solutions, i.e., they only exist when the perturbative quantumcorrections are incorporated into the action, and thus they are called Type-IIA quantum black holes.Of course, we still have to solve the H i , i = 1 , · · · , n v , as functions of the radialcoordinate τ . Since in this letter we are interested only in supersymmetric solutions, weautomatically know that [45–47] H i = a i − p i √ τ , r = 0 , (B.4)that is, the H i , i = 1 , · · · , n v , are given by harmonic functions on R . For supersymmetricsolutions we have to take r → and therefore the general metric (A.11) simplifies to ds = − e U ( τ ) dt + e − U ( τ ) δ mn dx m dx n , (B.5)where δ is the Euclidean metric on R . The near horizon τ → −∞ limit of the metric (B.5)is given by lim τ →−∞ ds = − c ) / (cid:12)(cid:12)(cid:12) κ ijk p i p j p k (cid:12)(cid:12)(cid:12) − / τ − dt + (3! c ) / (cid:12)(cid:12)(cid:12) κ ijk p i p j p k (cid:12)(cid:12)(cid:12) / τ δ mn dx m dx n . (B.6)The entropy of the Type-IIA quantum black holes is given by S = (3! c ) / π (cid:12)(cid:12)(cid:12) κ ijk p i p j p k (cid:12)(cid:12)(cid:12) / . (B.7)As we explained in section A, the connection one-forms A Λ can be explicitly obtained usingequation (A.13). For Type-IIA quantum black holes we obtain R = 0 , I i = I i = 0 , R ij = 0 , (B.8)which in turn implies that the following components of M MN are zero M i = M i = 0 , M = M = M ij = M ij = 0 , M i = M i = 0 . (B.9)From equations (A.13) and (B.1) we obtain– 17 – M = 12 (cid:90) e U M Mi p i dτ , (B.10)and since Ψ M = (cid:0) A Λ t , − A Λ t (cid:1) T , (B.11)we conclude using (B.9) that the only non-zero components of Ψ M are A i t = 12 (cid:90) e U M ij p j dτ , A t = − (cid:90) e U M j p j dτ . (B.12)This implies that the connection one-forms A i have only magnetic components, which giverise to the magnetic charges p i of the black hole solution (A.12). Notice however that thetime component of graviphoton A is non-zero, although the corresponding charges are zero:the magnetic one because i ∗ F is identically zero and the electric one thanks to a precisecancellation in the corresponding formula for q : q = (cid:90) S ∞ i ∗ G . (B.13) References [1] A. Strominger and C. Vafa,
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