A microscopic accounting of the entropy of a generic 5D supersymmetric
rotating black hole, arising from wrapped M2-branes in Calabi-Yau compactified
M-theory, is an outstanding unsolved problem. In this paper we consider an
expansion around the zero-entropy, zero-temperature, maximally rotating ground
state for which the angular momentum J_L and graviphoton charge Q are related
by J_L^2=Q^3. At J_L=0 the near horizon geometry is AdS_2 x S^3. As J_L^2 goes
to Q^3 it becomes a singular quotient of AdS_3 x S^2: more precisely, a
quotient of the near horizon geometry of an M5 wrapped on a 4-cycle whose
self-intersection is the 2-cycle associated to the wrapped-M2 black hole. The
singularity of the AdS_3 quotient is identified as the usual one associated to
the zero-temperature limit, suggesting that the (0,4) wrapped-M5 CFT is dual
near maximality to the wrapped-M2 black hole. As evidence for this, the
microscopic (0,4) CFT entropy and the macroscopic rotating black hole entropy
are found to agree to leading order away from maximality.Comment: 10 pages, no figure

Andrew Strominger

C. Vafa

D. Gaiotto

E. Verlinde

G.W. Gibbons

J. de Boer

J.M. Maldacena

L. Dyson

Monica Guica

N. Alonso-Alberca

Journal of High Energy Physics

English

arXiv

A microscopic accounting of the entropy of a generic 5D supersymmetric rotating black hole, arising from wrapped \(M2\)-branes in Calabi-Yau compactified \(M\)-theory, is an outstanding unsolved problem. In this paper we consider an expansion around the zero-entropy, zero-temperature, maximally rotating ground state for which the angular momentum \(J_L\) and graviphoton charge \(Q\) are related by \({J_L}^2 = Q^3\). At \(J_L = 0\) the near horizon geometry is \(AdS_2 × S^3\). As \({J_L}^2 → Q^3\) it becomes a singular quotient of \(AdS_3 × S^2\): more precisely, a quotient of the near horizon geometry of an \(M5\) wrapped on a 4-cycle whose self-intersection is the 2-cycle associated to the wrapped-\(M2\) black hole. The singularity of the \(AdS_3\) quotient is identified as the usual one associated to the zero-temperature limit, suggesting that the (0,4) wrapped-\(M5\) CFT is dual near maximality to the wrapped-\(M2\) black hole. As evidence for this, the microscopic (0,4) CFT entropy and the macroscopic rotating black hole entropy are found to agree to leading order away from maximality.Physic

Guica, Monica

Strominger, Andrew E.

Harvard University - DASH 

 Wrapped \(M2/M5\) Duality  (Article begins on next page)The Harvard community has made this article openly available.Please share how this access benefits you. Your story matters.Citation Guica, Monica, and Andrew E. Strominger. 2009. Wrapped\(M2/M5\) duality. Journal of High Energy Physics 2009(10): 36.Published Version doi:10.1088/1126-6708/2009/10/036Accessed February 19, 2015 9:23:15 AM ESTCitable Link http://nrs.harvard.edu/urn-3:HUL.InstRepos:8156532Terms of Use This article was downloaded from Harvard University's DASHrepository, and is made available under the terms and conditionsapplicable to Open Access Policy Articles, as set forth athttp://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAParXiv:hep-th/0701011v1  1 Jan 2007Wrapped M2/M5 DualityMonica Guica and Andrew StromingerJeﬀerson Physical Laboratory, Harvard University, Cambridge, MA 02138AbstractA microscopic accounting of the entropy of a generic 5D supersymmetric rotatingblack hole, arising from wrapped M2-branes in Calabi-Yau compactiﬁed M-theory, isan outstanding unsolved problem. In this paper we consider an expansion around thezero-entropy, zero-temperature, maximally rotating ground state for which the angularmomentum JL and graviphoton charge Q are related by J2L = Q3. At JL = 0 the nearhorizon geometry is AdS2 × S3. As J2L → Q3 it becomes a singular quotient of AdS3 ×S2: more precisely, a quotient of the near horizon geometry of an M5 wrapped on a4-cycle whose self-intersection is the 2-cycle associated to the wrapped-M2 black hole.The singularity of the AdS3 quotient is identiﬁed as the usual one associated to the zero-temperature limit, suggesting that the (0,4) wrapped-M5 CFT is dual near maximalityto the wrapped-M2 black hole. As evidence for this, the microscopic (0,4) CFT entropyand the macroscopic rotating black hole entropy are found to agree to leading order awayfrom maximality.Contents1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Near-maximal entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.1. BMPV→ AdS3 × S2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2. Near-maximal BMPV entropy . . . . . . . . . . . . . . . . . . . . . . . . 62.3. Far from maximality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73. General D6 charge with a single U(1) . . . . . . . . . . . . . . . . . . . . . . . 7Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91. IntroductionThe microscopic origin of the entropy of rotating 5D BPS black holes - known asBMPV black holes - in N = 8 and N = 4 theories has been understood for some time [1,2].In contrast, the origin of the entropy in 5D N = 2 theories, in particular those arisingfrom wrapped M2-branes in Calabi-Yau compactiﬁed M-theory, has remained enigmatic(except in some special cases [3]). In four dimensions, as long as the D6 charge p0 vanishes,the entropy has been understood in terms of the microstates of a certain (0,4) CFT - knownas the MSW CFT [4] - that arises from a wrapped M5-brane. However, when p0  = 0,the problem again remains unsolved. Due to the 4D-5D connection [5], the unsolved 5Dproblem is equivalent to the unsolved 4D case with p0 = 1.Naively, one might expect the SU(2)L angular momentum JL to be carried by excita-tions of the 5D black hole as in the N = 4,8 cases1 [1,2] and try to understand the JL = 0ground state by ﬁnding a dual of the near horizon AdS2 × S3. However the area-entropylawSBH = 2πqQ3 − J2L (1.1)where Q is the graviphoton charge, suggests that the ground state of the system is atmaximality, where J2L = Q3. The behaviour of the near maximal-solutions has been studiedin [6-8]. Despite the fact that the area goes to zero, the geometry remains smooth, while aU(1)L orbit of the horizon becomes null. Locally, the geometry becomes that of AdS3×S2[7], with curvatures and ﬂuxes corresponding to an M5-brane wrapped on a particular4-cycle in the Calabi-Yau. The particular 4-cycle is the one whose self-intersection is the1 In these cases the angular momentum JL is carried by left moving fermions of the D1 − D5CFT.12-cycle wrapped by the M2-brane carrying the black hole charges 2. Globally, the near-maximal BMPV is a quotient of AdS3×S2. At maximality, the orbits of the quotient havezero proper length, which looks a bit singular. However, exactly this type of singularityhas appeared before [10]. The quotient of AdS3 by the SL(2,IR)L element which takesw+ → e4π2TLw+ has zero length orbits as TL → 0. This quotient is known to be dual to amixed state of the boundary CFT in which the left movers are at temperature TL.This all suggests that the dual of the BMPV black hole is related to the MSW CFTwhich sits at the boundary of the limiting AdS3 × S2, with a temperature for left-moversdetermined by the deviation from maximality via the limiting form of the quotient. Wewill show in section 2 that this indeed produces the correct expression (1.1) for the entropyto leading order away from maximality.Expanding the geometry to second order away from maximality, we ﬁnd a deformationof AdS3×S2 in which the U(1) ﬁber associated to the graviphoton mixes with the spacetimedirections. We have not succeeded in interpreting this, but it suggests that the exact dualtheory beyond leading order away from maximality is some kind of deformation of theMSW CFT. An understanding of the general case would be of great interest.In section 3 we consider some Zp0 quotients of BMPV which arise as the 5D lift ofD6 charged 4D black holes. For special values of the charges we ﬁnd that these can bemapped to quotients of AdS3 × S2 even far from maximality, and that the microscopicentropy arising from the associated MSW CFT agrees with the Bekenstein-Hawking result.Some relevant AdS3 coordinate transformations are recorded in an appendix.2. Near-maximal entropy2.1. BMPV→ AdS3 × S2In this section we show that the total space of the graviphoton ﬁber bundle over BMPVis locally the same as the total space of the graviphoton ﬁber bundle over AdS3 × S2 fora certain relation between the graviphoton ﬂuxes, and that both are globally quotients ofAdS3 × S3. We ﬁrst consider the case of a single U(1) gauge ﬁeld and then generalize toN gauge ﬁelds.2 Not every 2-cycle is the self-intersection of an integral 4-cycle. In the following we restrictour attention to those 2-cycles for which this is the case. A dual relation - probably related to theone discussed herein - between such M2 and M5 branes was discussed in [9].2The metric for the near-horizon BMPV is3 (in Poincar´ e coordinates)ds25 =Q4￿−(cosBdtσ+ sinBσ3)2 +dσ2σ2 + σ21 + σ22 + σ23￿(2.1)where the right-invariant one-forms areσ1 = −sinψdθ + cosψ sinθdφσ2 = cosψdθ + sinψ sinθdφσ3 = dψ + cosθdφ(2.2)ψ is identiﬁed modulo 4π and the parameter B is related to the angular momentum JL bysinB =JLQ3/2 (2.3)At maximal rotation, sin2 B = 1 and J2L = Q3. The graviphoton ﬁeld strength in thecoordinates (2.1) is4F = dA, A =√Q2(cosBdtσ+ sinBσ3) (2.4)• One U(1)The total space of the U(1) bundle over BMPV deﬁnes a six manifold with metricds26 =Q4￿−(cosBdtσ+ sinBσ3)2 +dσ2σ2 + σ21 + σ22 + σ23￿+ (dy + A)2=Q4￿4Qdy2 +4√Qdy(cosBdtσ+ sinBσ3) +dσ2σ2 + σ21 + σ22 + σ23￿ (2.5)where y ∼ y + 2πn is the ﬁber coordinate. Deﬁningx =2cosB√Qy˜ y =√Q2ψ + sinBy(2.6)one ﬁndsds26 =Q4￿dx2 +2dxdtσ+dσ2σ2 + σ21 + σ22￿+ (d˜ y + ˜ A)2 (2.7)3 The t coordinate used here diﬀers by a factor of cosB from the usual asymptotic timecoordinate. We set ℓ5 = (4G5π )1/3 = 1.4 Our normalization is such that12πRF is integral.3with˜ A =√Q2sinθdφ (2.8)This is precisely the AdS3 × S2 near horizon metric for the charge p =√Q MSW string,provided p is an integer. Hence we see that a coordinate transformation which mixes upthe base and ﬁber transforms the near horizon geometry of p2 wrapped M2-branes to thatof p wrapped M5-branes.• N U(1)sNow let us repeat the discussion for the case of N −1 vector multiplets and associatedblack hole charges qA. We deﬁne pA as the 4-cycle whose self-intersection is qA, that isqA = 3DABCpBpC (2.9)The graviphoton charge is thenQ3/2 = DABCpApBpC (2.10)The attractor values of the vector moduli are tA = pA/√Q. We assume that qA are suchthat the pA are integers. (This is not the case for all integral qA). The metric is still givenby (2.1), but now the gauge ﬁelds areFA = dAA, AA =pA2(cosBdtσ+ sinB σ3) (2.11)The total space of the U(1)N bundle over BMPV has the metricds26 =Q4￿−(cosBdtσ+ sinBσ3)2 +dσ2σ2 + σ21 + σ22 + σ23￿+ (2tAtB − DAB)(dyA + AA)(dyB + AB)(2.12)whereDAB ≡ DABCtC , tA ≡ DABtB , yA ∼ yA + 2πnA, nA ∈ Z Z (2.13)Deﬁning˜ yA ≡ yA + (sinB − 1)tAtByB + 12pAψ (2.14)x =2cosB√QtAyA (2.15)4we can write the metric asds26 =Q4(dx2 +2dxdtσ+dσ2σ2 + σ21 + σ22) + (2tAtB − DAB)(d˜ yA + ˜ AA)(d˜ yB + ˜ AB)(2.16)with˜ AA = 12 pA cosθdφ (2.17)This is an N-dimensional ﬁbre bundle over AdS3 × S2, with the following identiﬁcations˜ yA ∼ ˜ yA + 2πnA + (sinB − 1)2πtAtCnC + 2πmpAx ∼ x +4π cosB√QtAnA (2.18)AdS3 × S3We have just seen that the total bundle space of near horizon wrapped M2’s andwrapped M5’s are equivalent. We now see that they are both quotients of AdS3×S3 witha ﬂat U(1)N−1 bundle5. Deﬁning˜ ψ = ψ +2sinB√QtAyA˜ σ3 = d ˜ ψ + cosθdφyAL = yA − tAtByB(2.19)and substituting in the BMPV metric with U(1)N bundle (2.12) we getds26 =Q4￿dx2 +2dxdtσ+dσ2σ2 + σ21 + σ22 + ˜ σ23￿− DABdyALdyBL (2.20)with the identiﬁcations˜ ψ ∼ ˜ ψ + 4πsinB√QtAnA + 4πmx ∼ x + 4πcosB√QtAnAyAL ∼ yAL + 2πnA − 2πtAtBnB(2.21)Of the six dimensions in (2.20), ﬁve are geometrical and the sixth is the U(1) ﬁber. Theﬁber U(1) is generated by the unit-normalized Killing vectorKF = tA ∂∂yA =2cosB√Q∂x +2sinB√Q∂ ˜ ψ (2.22)5 Closely related observations were made in [11].52.2. Near-maximal BMPV entropyTo get from AdS3×S3 in (2.20) to AdS3×S2, we regard the S3 factor as a U(1) bundleover S2 and reduce along ˜ ψ. To get to nonrotating BMPV, we regard the AdS3 factoras a U(1) bundle over AdS2 and reduce along x. For rotating BMPV, we reduce alongthe linear combination of x and ˜ ψ determined by the Killing vector (2.22). At maximalrotation, sinB = 1 and cosB = 0, so the reduction is entirely along ˜ ψ. This demonstratesthat maximal BMPV is locally AdS3 × S2.However, the relation between the generic BMPV and AdS3×S2 can not be so directbecause they have diﬀerent topologies. What happens is that the quotient (2.21) becomessingular in the maximal limit J2L → Q3, cosB → 0. This turns out to be exactly whatis needed: the singularity of the quotient is precisely of the form expected from the factthat the Hawking temperature TL (for the left movers of the MSW CFT) goes to zero atmaximality.To see this we write B = π2 −ǫ and deﬁne nA = ˆ nA −mpA. Keeping only the leadingterms in ǫ, (2.21) reduces to˜ yA ∼ ˜ yA + 2πˆ nAx ∼ x − 4πǫm(2.23)Writing the identiﬁcation this way it becomes clear that in the maximally-rotating limitthe ˆ nA identiﬁcation only acts on the N ﬁber coordinates, while the m identiﬁcation onlyacts on AdS3. Moreover, the AdS3 identiﬁcation lives purely in the SL(2,IR)L factor of theisometry group. AdS3 quotients of this type were analyzed in [10]. While t is a standardnull coordinate on the Poincar´ e diamond of the boundary of AdS3, x in fact turns outto be a Rindler coordinate. The SL(2,IR)L-invariant AdS3 vacuum then gives, after xidentiﬁcations, a thermal state for the left movers of the boundary CFT ( in this case theMSW CFT). This quotient produces a boundary CFT in a thermal ensemble with rightmoving temperature TR = 0 and left moving temperature equal to the magnitude of the xshift divided by 4π2. Hence from (2.23) we haveTL =ǫπ(2.24)As expected, TL → 0 at maximality and the quotient becomes singular. The entropy ofthe MSW string at temperature TL isSmicro =π2cLTL3= 2πDABCpApBpCǫ = 2πQ3/2ǫ (2.25)6The Bekenstein-Hawking entropy of BMPV isSmacro = 2πQ3/2 cosB (2.26)which agrees with (2.25) near maximality. This supports the conjecture that the excitationsof the BMPV black hole near its maximally spinning ground state are dual to to the MSWstring on the boundary of the limiting AdS3 × S2 × CY attractor.2.3. Far from maximalityIn the above derivation, we made crucial use of the fact that the black hole was nearmaximal rotation and that the discrete identiﬁcations could be approximated by the leadingterms in the near-maximal expansion parameterized by ǫ = π2 − B. In this approximationthe resulting geometry has a simple interpretation as a well-studied quotient of AdS3.However at next to leading order, the identiﬁcations mix up the base and ﬁber coordinatesin a way that makes the spacetime interpretation obscure. It is quite possible that thereis some way of making sense of the general case, but we have not succeeded in doing so.This indicates that, while the MSW string provides a good description of near-maximalexcitations, it is not the exact dual of BMPV. Perhaps the dual is some kind of deformationof MSW.It turns out that one can do a bit better - for the case of a single U(1) - by consideringquotients of BMPV. This will be the subject of the next section.3. General D6 charge with a single U(1)In the preceding section we argued that the BPS excitations of a BMPV black holevery near its maximally rotating ground state were described by the dual MSW string.Using the 4D-5D connection [5] these become excitations of the maximally D0-charged4D black hole with D6 charge p0 = 1. In this section we will consider general p0, but asingle U(1). We will ﬁnd -for certain charges - evidence for a general relation to the MSWstring not limited to near maximality. We ﬁnd it clearest to start with the near horizonM5 geometry known to be dual to MSW, and show that it can be recast as a quotient ofBMPV.7We start with a well-understood quotient of AdS3 ×S2 (equation (2.7)) accompaniedby a twist in the graviphoton connection, which can generically be written asx ∼ x + 4π2TLk˜ y ∼ ˜ y + 2πw + 2πkθ(3.1)θ is a chemical potential for graviphoton charge. We then use the bundle coordinate trans-formations (2.14),(2.15) to local BMPV bundle coordinates and ask for which values of TLand θ this system actually describes BMPV. Of course, bundle coordinate transformationswhich mix up the base and the ﬁber are not known in general to be a symmetry of stringtheory (unless of course the U(1) is a Kaluza-Klein U(1)), so it is not guaranteed thatthis will give a sensible result. Hence our ﬁnal result, which does appear sensible, must beregarded as a conjecture.6 The identiﬁcations (3.1) in BMPV coordinates becomey ∼ y +2π2pTLkcosBψ ∼ ψ +4πwp+ 4πk(θp− πTL tanB)(3.2)where p2 = Q. In order for the graviphoton charge quantization condition to be the onedictated by M theory, the y ﬁber must be identiﬁed modulo 2πk, which requiresTL =cosBπp(3.3)In order for y to be a true ﬁber over BMPV, the y identiﬁcations generated by k must notinvolve a shift in the base. This requiresθ = sinB. (3.4)We are then left with the identiﬁcationsy ∼ y + 2πk, ψ ∼ ψ +4πwp(3.5)This contrasts with usual BMPV for which one has ψ ∼ ψ + 4πw. However (3.5) has asimple interpretation. The factor of p in (3.5) means that we have BMPV with an S3/Zphorizon.6 We note that in the near maximal discussion of the previous section, the coordinate trans-formations did not, at leading order, mix the ﬁber and base.8The Zp quotient is the near horizon geometry of the 5D lift of a 4D black hole withD6 charge p0 = p and q0 = 2JL. This has Bekenstein Hawking entropySBH = 2πQ3/2 cosBp0 = 2πp2 cosB (3.6)On the other hand this was related to the MSW string at temperature (3.3). This has anentropySmicro =π23cLTL = 2πp2 cosB = SBH (3.7)This motivates the conjectured duality between the 4D black hole with charges (p0,p1,q1,q0) =(p,0,p,q0) and the MSW CFT of charge p M5-branes at temperatureTL =1πps1 −q204p2 (3.8).AcknowledgementsWe are grateful to Davide Gaiotto, Lisa Huang, Wei Li, Erik Verlinde and Xi Yin forhelpful discussions. This work has been supported in part by the DOE grant DE-FG02-91ER40654.Appendix A.We list herein the transformations from the AdS3 coordinates in (2.7) to the Poincar´ ecoordinates used in [10]. In Poincar´ e coordinates, the metric readsds2 = Qdw+dw− + dξ2ξ2 +Q4(σ21 + σ22 + ˜ σ23) (A.1)The transformation from (2.7) to Poincar´ e isw+ = exw− = 12(t − σ)ξ2 = σex(A.2)9References[1] A. Strominger, C. Vafa, “Microscopic origin of the Bekenstein-Hawking entropy”Phys.Lett. B 379 (1996), 99-104 [arXiv: hep-th/9601029].[2] J.C. Breckenridge, R.C. Myers, A.W. Peet, C. Vafa, “D-branes and spinning blackholes” Phys.Lett.B 391 (1997), 93-98 [arXiv: hep-th/9602065].[3] C. Vafa, “Black holes and Calabi-Yau threefolds” Adv.Theor.Math.Phys. 2 (1998)207-218 [arXiv: hep-th/9711067].[4] J. M. Maldacena, A. Strominger, E. Witten, “Black hole entropy in M-theory” JHEP9712, 002 (1997) [arXiv:hep-th/9711053].[5] D. Gaiotto, A. Strominger, X. Yin, “New connections between 4-D and 5-D blackholes” JHEP 0602 (2006) 024 [arXiv: hep-th/0503217].[6] G.W. Gibbons, C.A.R. Herdeiro, “Supersymmetric rotating black holes and causalityviolation” Class.Quant.Grav. 16 (1999) 3619-3652 [arXiv: hep-th/9906098].[7] N. Alonso-Alberca, E. Lozano-Tellechea, T. Ortin, “The near horizon limit of theextreme rotating D = 5 black hole as a homogeneous space-time” Class.Quant.Grav.20 (2003) 423-430 [arXiv: hep-th/0209069].[8] L. Dyson, “Studies Of The Over-Rotating BMPV Solution” [arXiv: hep-th/0608137].[9] J. de Boer, M.C.N. Cheng, R. Dijkgraaf, J. Manschot, E. Verlinde, “A Farey Tailfor Attractor Black Holes” JHEP 0611 (2006) 024 [arXiv: hep-th/0608059], and talkgiven by E. Verlinde at Strings ’06.[10] J.M. Maldacena, A. Strominger, “AdS(3) black holes and a stringy exclusion principle”JHEP 9812 (1998) 005 [arXiv: hep-th/9804085].[11] M. Cvetic, F. Larsen, “Near horizon geometry of rotating black holes in ﬁve-dimensions” Nucl.Phys.B 531 (1998) 239-255 [arXiv: hep-th/9805097].10

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Wrapped M2/M5 Duality

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