Exact microstate counting for dyonic black holes in AdS4
aa r X i v : . [ h e p - t h ] A ug SISSA 41/2016/FISI
Exact microstate counting for dyonic black holes in AdS Francesco Benini,
1, 2
Kiril Hristov, and Alberto Zaffaroni
4, 5 SISSA, via Bonomea 265, 34136 Trieste, Italy — INFN, Sezione di Trieste Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom INRNE, Bulgarian Academy of Sciences, Tsarigradsko Chaussee 72, 1784 Sofia, Bulgaria Dipartimento di Fisica, Universit`a di Milano-Bicocca, I-20126 Milano, Italy INFN, sezione di Milano-Bicocca, I-20126 Milano, Italy (Dated: August 29, 2016)We present a counting of microstates of a class of dyonic BPS black holes in AdS which preciselyreproduces their Bekenstein-Hawking entropy. The counting is performed in the dual boundarydescription, that provides a non-perturbative definition of quantum gravity, in terms of a twistedand mass-deformed ABJM theory. We evaluate its twisted index and propose an extremizationprinciple to extract the entropy, which reproduces the attractor mechanism in gauged supergravity. Supersymmetric black holes in string theory constituteimportant models to test fundamental questions aboutquantum gravity in a relatively simple setting. The mainquestion we would like to address here is the origin of theblack hole (BH) entropy, which statistically is expectedto count the number of degenerate BH configurations.String theory provides a microscopic explanation for theentropy of a class of asymptotically flat black holes [1].Much less is known about asymptotically AdS ones infour or more dimensions.In principle AdS/CFT [2] provides a non-perturbativedefinition of quantum gravity in asymptotically AdSspace, as a dual boundary quantum field theory (QFT).The BH microstates appear as particular states in theboundary description. The difficulty with this approachis the need to perform computations in a strongly coupledQFT, but the development of exact non-perturbativetechniques makes progress possible. We recently reported[3] on a particular example of magnetically charged BPSblack holes in AdS [4] with a known field theory dual—topologically twisted ABJM theory. Using the techniqueof supersymmetric localization we were able to calcu-late in an independent way the (regularized) number ofground states of the theory and successfully match it withthe leading macroscopic entropy of the black holes.In this Letter we discuss a function Z ( u a ) that encodesthe quantum entropies of static dyonic BPS black holes inAdS , computed non-perturbatively from the dual QFTdescription, and show that its leading behavior repro-duces the Bekenstein-Hawking entropy [5].In particular we show that, at leading order, the en-tropy of BPS black holes with magnetic charges p a , elec-tric charges q a and asymptotic to AdS × S can be ob-tained by extremizing the quantity I = log Z ( u a ) − i X a u a q a (1)with respect to a set of complexified chemical potentials u a for the global U (1) flavor symmetries of the boundarytheory. Z ( u a ) is the topologically twisted index [6] ofthe ABJM theory [7] which explicitly depends on the magnetic charges p a (see [8] for other examples). Theentropy is given by S = I (ˆ u ) evaluated at the extremum,with a constraint on the charges that S be real positive.As we will see, the extremization of I is equivalent tothe attractor mechanism for AdS black holes in gaugedsupergravity. We also argue, generalizing [3], that the ex-tremization of I selects the exact R-symmetry of the su-perconformal quantum mechanics dual to the AdS hori-zon region. We notice strong similarities between ourformalism and those based on Sen’s entropy functional[9] and the OSV conjecture [10]. The Black Holes .—We consider dyonic BPS BHs thatcan be embedded in M-theory and are asymptotic toAdS × S . They are more easily described as solutions inthe STU model, a four-dimensional N =2 gauged super-gravity with three vector multiplets, which is a consistenttruncation both of M-theory on S , and of the 4d max-imal N =8 SO (8) gauged supergravity [11]. The modelcontains four Abelian vector fields (one is the gravipho-ton) corresponding to the U (1) ⊂ SO (8) isometries of S .In 4d N =2 supergravities with n V vector multiplets,one can use the standard machinery of special geome-try [12]. The Lagrangian L of the theory is completelyspecified by the prepotential F ( X Λ ), which is a homo-geneous holomorphic function of sections X Λ , and thevector of Fayet-Iliopoulos (FI) terms G = ( g Λ , g Λ ). Thesymplectic index Λ = 0 , , . . . , n V runs over the gravipho-ton and the n V vectors in vector multiplets. The scalars z i in vector multiplets, with i = 1 , . . . , n V , parametrizea special K¨ahler manifold M and X Λ are sections of asymplectic Hodge vector bundle on M . The formalism iscovariant with respect to symplectic Sp (2 n V + 2) trans-formations. Indicating as ( A Λ , A Λ ) the 2 n V + 2 com-ponents of a symplectic vector A , the scalar product is h A, B i = A Λ B Λ − A Λ B Λ . One defines the covariantly-holomorphic sections V = e K ( z, ¯ z ) / (cid:18) X Λ ( z ) F Λ ( z ) (cid:19) (2)on M , where K is the K¨ahler potential and F Λ ≡ ∂ Λ F .They satisfy D ¯ ı V ≡ (cid:0) ∂ ¯ ı − ∂ ¯ ı K (cid:1) V = 0. The K¨ahlerpotential is then determined by hV , Vi = − i .The ansatz for dyonic black holes is of the form ds = − e U ( r ) dt + e − U ( r ) (cid:0) dr + V ( r ) ds g (cid:1) (3)where Σ g is a Riemann surface of genus g , and the scalarfields z i are assumed to only have radial dependence. Wecan write the metric on Σ g locally as ds g = dθ + f κ ( θ ) dϕ , f κ ( θ ) = sin θ κ = 1 θ κ = 0sinh θ κ = − κ = 1 for S , κ = 0 for T , and κ = − g with g >
1. The scalar curvature is 2 κ and the volume isVol(Σ g ) = 2 πη , η = (cid:26) | g − | for g = 11 for g = 1 . (5)The magnetic and electric charges of the black hole are Z Σ g F Λ = Vol(Σ g ) p Λ , Z Σ g G Λ = Vol(Σ g ) q Λ , (6)where G Λ = 8 πG N δ ( L d vol ) /δF Λ and G N is the New-ton constant. This particular normalization ensures thatthe BPS equations are independent of g (besides a lin-ear constraint). The charges are collected in the vector Q = ( p Λ , q Λ ). The vector G of FI terms controls the gaug-ing and determines the charges of the gravitini under thegauge fields. In a frame with purely electric gauging g Λ ,the lattice of electro-magnetic charges is η g Λ p Λ ∈ Z , η G N g Λ q Λ ∈ Z (7)not summed over Λ. It turns out that the BPS equationsfix the more stringent condition hG , Qi = − κ , (8)that we call the linear constraint.It has been noticed in [13] that the BPS equations ofgauged supergravity for the near-horizon geometry canbe put in the form of “attractor equations”. One definesthe central charge of the black hole Z and the superpo-tential L : Z = hQ , Vi = e K / (cid:0) q Λ X Λ − p Λ F Λ (cid:1) L = hG , Vi = e K / (cid:0) g Λ X Λ − g Λ F Λ (cid:1) . (9)The BPS equations for the near-horizon geometry ds = − r R A dt + R A r dr + R S ds g (10)with constant scalar fields z i imply the following twoequations [13]: Z − i R S L = 0 and D j (cid:0) Z − i R S L (cid:1) = 0, where D j = ∂ j + ∂ j K , besides hG , Qi = − κ . Theseequations can be rewritten as ∂ j ZL = 0 , − i ZL = R S . (11)In other words, the scalars z i at the horizon take a valuesuch that the quantity − i Z / L has a critical point on M and then its value is proportional to the Bekenstein-Hawking black hole entropy.Notice that a condition to have BHs with smooth hori-zon is that − i Z / L be real positive at the critical point.Since the critical-point equations already fix the values ofthe scalars, this condition becomes a second (non-linear)constraint on the charges. Therefore the domain of al-lowed electro-magnetic charges has real dimension 2 n V (before imposing quantization). There are other inequal-ities to be satisfied by the charges, for instance to ensurethat also R A be positive.In the case of very special K¨ahler geometry, i.e. thatthe prepotential takes the form F = d ijk X i X j X k /X orsymplectic transformations thereof, general solutions tothe near-horizon BPS equations as well as full BH solu-tions have been found in [14]. That analysis guaranteesthat all near-horizon solutions can be completed into fullBH solutions.Our focus is on the STU model, which has n V = 3 andprepotential F = − i √ X X X X , (12)with purely electric gauging g Λ ≡ g , g Λ = 0. Then theAdS vacuum has radius L = 1 / g . Note that all dy-onic BH solutions with electric charges have complex pro-files for the scalars, i.e. the axions are turned on. The dual field theory. —M-theory on AdS × S has adual holographic description as a three-dimensional su-persymmetric gauge theory, the ABJM theory [7], whichprovides a non-perturbative definition thereof. In N =2notation, the ABJM theory is a U ( N ) × U ( N ) − Chern-Simons theory (the subscripts are the levels) with bi-fundamental chiral multiplets A i and B j , i, j = 1 , N, N ) and (
N , N ) representationsof the gauge group, respectively, and with superpotential W = ε ik ε jl Tr A i B j A k B l . The theory has N =8 super-conformal symmetry and SO (8) R-symmetry. The iden-tification between gravitational and QFT parameters is L G N = 12 g G N = 2 √ N / . (13)The “topologically twisted index” of an N =2 three-dimensional theory is its supersymmetric Euclidean par-tition function on S × S with a topological twist on S [6]. Its higher-genus generalization, namely the twistedpartition function on Σ g × S , has been constructed aswell [15, 16]. They depend on a set of integer magneticfluxes p a and complex fugacities y a , along the Cartangenerators of the flavor symmetry group.In the present case, to make the enhanced symmetrymore manifest, we introduce an index a = 1 , , , U (1) ⊂ SO (8). This isdone by introducing a basis of four R-symmetries R a ,each acting with charge 2 on one of the chiral fields andzero on the others. Then the magnetic fluxes identifya U (1) subgroup of SO (8) used to twist, and are re-quired by supersymmetry to satisfy P a p a = 2 g −
2. Thecomplex fugacities y a = exp iu a must satisfy Q a y a = 1( P a u a ∈ π Z ) and encode background values for theflavor symmetries. Writing u a = ∆ a + iβσ a (where β is the length of S ), we can identify ∆ a with flavor flatconnections and σ a with real masses.The Hamiltonian definition of the index is [6] Z ( u a , p a ) = Tr ( − F e i P a =1 ∆ a J a e − βH , (14)where J a = ( R a − R ) are the three independent fla-vor symmetries and H is the twisted Hamiltonian on S , explicitly dependent on the magnetic charges p a andthe real masses σ a . Due to the supersymmetry algebra Q = H − P a =1 σ a J a , the index Z ( u a , p a ) is a mero-morphic function of y a . For simplicity, we will keep thedependence on p a implicit and use the shorthand nota-tion σJ = P a =1 σ a J a . We stress that, in general, (14)is well-defined only for complex u a while the index for σ a = 0 is defined by analytic continuation. We wouldlike to see how we can extract the BH entropies from Z . Statistical interpretation. —The partition function Z ( u ) describes a supersymmetric ensemble which iscanonical with respect to the magnetic charges ( i.e. allstates have the same, fixed, magnetic charges) but grandcanonical with respect to the electric charges ( i.e. it isa sum over all electric charge sectors, with fixed chemi-cal potentials u a ). A similar viewpoint in BH physics isadvocated in [17]. We can decompose Z as a sum oversectors with fixed charges q a under R a / J a , R a ∈ Z , up to a possiblezero-point shift in the vacuum): Z ( u ) = X q e i P a =1 u a ( q a − q ) Z q . (15)We would like to identify S q ≡ R e log Z q with the leadingentropy of a BH of fixed electric charges q a . We take thereal part to remove the effect of a possible overall sign.An important assumption is that ( − F in the trace (14)does not cause dangerous cancelations at leading order.We can Fourier transform the previous expression withrespect to the three independent ∆ a to obtain X ′ q Z q = Z d ∆ a (2 π ) e − i P b =1 ∆ b ( q b − q ) Z ( u ) , (16)where prime means that the sum is taken at fixed integer q a − q . As we will see, for supergravity BHs both the electric charges q a and log Z are of order N / , thereforethe previous expression can be evaluated at large N usinga saddle point approximation: X ′ q Z q = exp h log Z (ˆ u ) − i X a =1 ˆ u a ( q a − q ) i (17)at leading order, where ˆ u a is a solution for u a to ∂∂u a h log Z ( u ) − i X b =1 u b ( q b − q ) i = 0 (18)with a = 1 , ,
3. This saddle point in general gives com-plex values for ˆ u a . The sum on the LHS of (17) will alsobe dominated by a specific value of q , corresponding tothe electric R-charge of the black hole. For that value: S q = R e h log Z (ˆ u ) − i X a =1 ˆ u a ( q a − q ) i . (19)We can restore the permutation symmetry between thecharges, part of the Weyl group of SO (8), by introducing I ( u ) ≡ log Z ( u ) − i X a =1 u a q a . (20)Eqn. (18) is equivalent to extremization of I and theentropy is given by S q = R e I (ˆ u ).This argument does not determine the R-charge of theBH, essentially because the index Z ( u ) lacks a chemicalpotential for it. However from the attractor equations(11) it follows that, for given magnetic charges p a andflavor electric charges q a − q , there is at most one valueof q leading to a large smooth BH. Our argument thengives an unambiguous prediction for the leading entropyof that BH. RG flow interpretation. —We can extract more infor-mation from the index if we interpret the BH as an holo-graphic RG flow. The near-horizon geometry of BPSblack holes contains an AdS factor permeated by con-stant electric flux, where the super-isometry algebra isenhanced to su (1 , | S to an ensemble of su (1 , | sl (2 , R ) × u (1) c wherethe second factor is the IR superconformal R-symmetry,which is some linear combination of U (1) ⊂ SO (8). Inthe near-horizon canonical ensemble this implies that allBH states have zero U (1) c charge (by an argument sim-ilar to that in [18]). We will assume that there are noother contributions outside the horizon.The asymptotic behavior of electrically charged BHsolutions with axions turned on suggests that the dualABJM theory is also deformed by real masses σ a . In gen-eral, they lift a possible vacuum degeneracy of the Hamil-tonian H . The presence of AdS with constant electricflux, though, indicates that there should be a large vac-uum degeneracy for a modified Hamiltonian H nh in whichthe energy of states gets an extra contribution linear inthe charge: H nh ( σ ) = H ( σ ) − σ . From the supersymme-try algebra Q = H nh we conclude that H nh ≥
0, andthe index gets contribution only from its ground states.We can rewrite the index in (14) as Z (∆ , σ ) = Tr ′ e iπR trial (∆) e − βσJ , (21)where Tr ′ = Tr H nh =0 . We introduced a trial R-current R trial (∆) ≡ R + ∆ J/π that parametrizes the mixing ofthe R-symmetry with the flavor symmetries, with R areference R-symmetry such that e iπR = ( − F .We want to argue, generalizing [3], that the supercon-formal R-symmetry R c of the Hamiltonian H nh can befound by extremizing Z (∆ , σ ) for fixed values of σ a . Letˆ∆ a be the value such that R trial ( ˆ∆) = R c . One com-putes ∂ log Z/∂ ∆ a (cid:12)(cid:12) ˆ∆ a = i h J a e − βσJ i / h e − βσJ i , using thatat zero temperature the density matrix is uniformly dis-tributed over the ground states of H nh , and that R c = 0in those states as argued above. The expression on theright is imaginary, implying that ˆ∆ a are determined byextremizing the index with respect to ∆ a at fixed σ a : ∂ R e log Z (∆ , σ ) ∂ ∆ a (cid:12)(cid:12)(cid:12) ˆ∆ = 0 . (22)This is the generalization of the I -extremization princi-ple proposed in [3]. Assuming the large N factorization h Je − βσJ i = h J ih e − βσJ i , we also have ∂ I m log Z (∆ , σ ) ∂ ∆ a (cid:12)(cid:12)(cid:12) ˆ∆ = i h J a i ≡ i ( q a − q ) , (23)where h J a i is the charge of the vacuum density matrix.This determines the relation between the flavor charges q a − q and σ a . Since Z (∆ , σ ) is a holomorphic functionof u a = ∆ a + iβσ a , we can summarize the result in thecomplex equation ∂ log Z ( u ) ∂u a (cid:12)(cid:12)(cid:12) ˆ u = i ( q a − q ) , (24)which determines both ˆ∆ a and σ a as functions of q a .From eqn. (21), at the critical point Z ( ˆ∆ , σ ) = e − βσ h J i Tr ′ e − P a =1 βσ a q a e S q . (25)The real part of the logarithm of this expression repro-duces the result of the statistical argument, namely S q = R e h log Z (ˆ u ) − i X a =1 ˆ u a q a i . (26)An advantage of this derivation is that we can argue, atleast at leading order, that e S q is the number of groundstates, without dangerous signs that could cause cance-lations.We can also write the entropy in a slightly differentform and make a conjecture for the value of the fourthcharge. Since ˆ u only depends on the differences q a − q and P a u a ∈ π Z , we can always shift the integer charges q a and write the entropy in the permutationally symmet-ric and holomorphic form S q = log Z (ˆ u ) − i X a =1 ˆ u a q a = I (ˆ u ) , (27)up to O ( N ) terms which are invisible in the large N limit. The determination of the logarithm is such thatlog Z is real for σ a = 0 and extended by continuity. Therequirement that (27) be real positive fixes the fourthcharge. Interestingly, this is precisely the constraint (11)that comes from supergravity. Explicit match for ABJM .—The large N expression forthe index of ABJM was found in [3, 15] for the case ofreal u a , and we can extend it to the complex plane usingholomorphy:log Z = N / √ u u u u X a =1 p a u a . (28)This is valid for P a u a = 2 π and 0 < R e u a < π . The I -extremization principle (24) is equivalent to the ex-tremization of I QFT = X a =1 (cid:18) N / √ u u u u p a u a − i q a u a (cid:19) . (29)Then the entropy is given by S q = I QFT (ˆ u ), with theconstraint on the charges that I QFT (ˆ u ) be positive.In supergravity, the BH entropy is determined by S BH = Area4 G N = − i ZL πη G N ≡ I SUGRA (30)using (12), and I SUGRA should be extremized with re-spect to X Λ . We can identify the index Λ = { , , , } with a = { , , , } , as well as 2 πX a / P b X b = u a sincethey have the same domain and constraint: I SUGRA = η gG N X a =1 (cid:18) √ u u u u p a u a − iq a u a (cid:19) . (31)Identifying the integers in (7) with the charges p a , q a ,respectively, and using (13) we obtain a perfect match I QFT = I SUGRA . The field theory extremization princi-ple corresponds to the supergravity attractor mechanism:they lead to the same entropy and non-linear constrainton the charges.We thank J. de Boer, A. Gnecchi, N. Halmagyi and S.Murthy for instructive clarifications. FB is supported bythe MIUR-SIR grant RBSI1471GJ. AZ is supported bythe MIUR-FIRB grant RBFR10QS5J. [1] A. Strominger and C. Vafa, Phys. Lett.
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