CFT Duals of Black Rings With Higher Derivative Terms
aa r X i v : . [ h e p - t h ] M a r WITS-CTP-79
CFT Duals of Black Rings With Higher Derivative Terms
Kevin Goldstein , Hesam Soltanpanahi National Institute for Theoretical Physics, GautengSchool of Physics and Center for Theoretical Physics,University of the Witwatersrand,WITS 2050, Johannesburg, South Africa
Abstract
We study possible CFT duals of supersymmetric five dimensional black rings in thepresence of supersymmetric higher derivative corrections to the N = 2 supergravityaction. A Virasoro algebra associated to an asymptotic symmetry group of solutionsis defined by using the Kerr/CFT approach. We find the central charge and computethe microscopic entropy which is in precise agreement with the macroscopic entropy.Although apparently related to a different aspect of the near-horizon geometry and adifferent Virasoro algebra, we find that the c-extremization method leads to the samecentral charge and microscopic entropy computed in the Kerr/CFT approach. Therelationship between these two point of view is clarified by relating the geometry toa self-dual orbifold of AdS . e-mail:kevin (at) neo.phys.wits.ac.za e-mail:hesam.soltanpanahisarabi (at) wits.ac.za ontents A decade before Maldacena’s AdS/CFT conjecture of gravity-gauge duality [1], a relation-ship between gravity on AdS and the two dimensional conformal group was found by Brownand Henneaux [2]. They found that appropriate boundary conditions, which preserve theasymptotically AdS geometry, generate two Virasoro algebras. It is not a great leap torelate this asymptotic symmetry group to some CFT living on the boundary and applyingthe Brown-Henneaux formalism to asymptotically AdS black hole solutions allows one torelate the macroscopic Bekenstein-Hawking entropy to the number of states of a boundaryCFT [3, 4, 5, 6]. One uses the fact that if the asymptotic symmetry group is a Virasoroalgebra with a particular central charge, the Cardy formula relates the central charge toentropy of the CFT in the large charge limit. We refer to the entropy obtained from theBrown-Henneaux formalism as the microscopic entropy but it should be recalled that, inmany cases, the existance of a microscopic CFT is only hypothesised.More recently, a clever extension of the Brown-Henneaux approach to the SL(2, R ) × U (1)near-horizon symmetry of the extremal Kerr black hole, led to the proposal of a Kerr/CFTcorrespondence [7]. It was found that one can find an asymptotic symmetry group, corre-sponding to the U (1) part of the isometry, with a Virasoro algebra whose central chargecorrectly accounts for the Bekenstein-Hawking entropy via the Cardy formula.The Kerr/CFT correspondence has been generalised to, and verified for higher dimen-sional rotating black holes [8, 9, 10, 11, 12, 13, 14, 15], Lagrangians with topological termsin four and five dimensions [16] and Lagrangians with higher derivative terms [17]. Notably,it was shown that in certain theories with gravity coupled to matter fields the associatedcentral charge coming from the gravitational degrees of freedom is sufficient to account forthe entropy [11, 16].Recently, people have tried to embed the Kerr/CFT approach in string theory [18, 19,20]. In light of this we were motivated to study possible CFT duals of five dimensionalsupersymmetric black rings [21] in the presence of special higher derivative terms which arethe supersymmetric completion of a mixed gauge-gravitational Chern-Simons term [22].These solutions have SL(2, R ) × U (1) × SO(3) near-horizon symmetries corresponding to anS fibred over AdS × S . For various technical reasons outlined below, this seems to be apromising case to study. Firstly, with the addition of higher derivative terms one might1ope to study corrections to the entropy beyond the Cardy limit. It turns out that thenear horizon geometry retains the same form upon addition of these higher derivative terms[23, 24, 25], making calculations tractable. Secondly, the rich near horizon geometry of theSUSY black rings allows for various approaches to finding the entropy to be studied andcompared. In particular, the S part, which carries the angular momentum of the blackring, is fibred over the AdS part such that locally one has an AdS geometry called aself-dual orbifold of AdS [26, 27]. So in addition to using a generalisation of Kerr/CFT,corresponded to the U (1) part of the isometry of near horizon geometry [8], the fact thatone locally has an AdS means that the c-extremisation formalism [28] can be applied. Thisapproach is based on the relationship between the conformal anomaly and the variation ofthe gravitational action with respect to a metric of the form AdS × Y where Y is compact.One obtains two Virasoro algebras corresponded to the SL (2 , R ) L × SL (2 , R ) R isometry ofAdS .We found that comparing the result of applying Kerr/CFT and c-extremisation illumi-nating. The fact that one only has a local AdS means an application of the c-extremisationformalism is not straight forward. The SL (2 , R ) L × SL (2 , R ) R isometry of AdS is reducedto SL (2 , R ) L × U (1). On the CFT side one expects that this kills the right-handed ex-citations, which means there is a (0,4)-CFT corresponded to SUSY black ring, and themicroscopic entropy is given by one part of the Cardy formula, [29, 30, 31, 32] S mic = r c L ˆ q . (1.1)On the other hand, in the Kerr/CFT approach, where the Virasoro is intimately relatedto the U (1) part of the isometry, which descends from the right-handed SL (2 , R ) R , onemay naively expect that the central charge obtained descends from the right-handed centralcharge of global AdS . This is not the case. From CFT point of view, we know that thereis a (0,4)-CFT corresponding to the SUSY black ring [29, 30, 31, 32] and we expect thatthe left central charge contributes to the microscopic entropy. We will show that bothc-extremization and Kerr/CFT approaches lead to the same central charge which is theleft central charge c L . This agreement was shown for supersymmetric black ring withouthigher derivative terms [8]. The equality of the two central charges when higher derivativecorrections are added is a non-trivial new result.The rest of this paper is organised as follows. In section 2 we briefly review supersym-metric black ring solutions of five dimensional superconformal gravity in the presence ofhigher derivative terms. Then we apply the Kerr/CFT approach to the supersymmetricblack ring in section 3. The associated central charge is computed and the agreement be-tween the microscopic entropy and macroscopic entropy is shown. Section 4 is devoted tothe application of the c-extremization formalism for the above black ring solution. We showthat the associated left central charge and microscopic entropy are in agreement with theKerr/CFT results. Finally in section 5, we will summarize and discuss our results. In this section we review five dimensional N = 2 supergravity with higher derivative cor-rections associated with a mixed gauge gravitational Chern-Simons term. We will do thatin the context of an off-shell formalism which involves superconformal gravity [22]. The2mportant feature of the formalism is that variation of fermionic fields does not dependexplicitly on the form of the action. In particular, the variation of fermionic fields does notchange if higher order derivative terms are added.Compactification of M-theory on a six dimensional Calabi-Yau manifold results in N = 2supergravity in five dimensions. In [22] it was shown that by enlarging the symmetries tosuperconformal gravity (by adding some auxiliary fields) one can find a generic form for thefermionic variations which leave the action invariant. As mentioned, this form is valid forany number of higher derivative terms. The bosonic action, up to 4th order, is given by I = 116 πG Z d x p | g | ( L + L ) , (2.1)where L = ∂ a A iα ∂ a A αi + (2 ν + A ) D ν − A ) R ν − A ) v ν I F Iab v ab + 14 ν IJ ( F Iab F J ab + 2 ∂ a X I ∂ a X J ) + e − C IJK ǫ abcde A Ia F Jbc F Kde , (2.2)is the tree level part of the action and L = c I (cid:18) e ǫ abcde A Ia R bcfg R defg + 18 X I C abcd C abcd + 112 X I D + 16 F Iab v ab D + 13 X I C abcd v ab v cd + 12 F Iab C abcd v cd + 83 X I v ab ˆ D b ˆ D c v ac + 43 X I ˆ D a v bc ˆ D a v bc + 43 X I ˆ D a v bc ˆ D b v ca − e X I ǫ abcde v ab v cd ˆ D f v ef + 23 e F Iab ǫ abcde v cf ˆ D f v de + e − F Iab ǫ abcde v cf ˆ D d v ef − F Iab v ac v cd v db − F Iab v ab v + 4 X I v ab v bc v cd v da − X I ( v ) (cid:19) , (2.3)are all four derivative terms which are related to the mixed gauge-gravitational Chern-Simons term c I A I ∧ R ∧ R by supersymmetry transformations [22]. In this action C IJK and c I are the intersection numbers and the second Chern class of internal space CY respectively, A = A iα A αi , v = v ab v ab and ν = 16 C IJK X I X J X K , ν I = 12 C IJK X J X K , ν IJ = C IJK X K . (2.4)The fields appearing in the action are arranged in Weyl, vector and hyper multiplets. TheWeyl multiplet contains the metric, a 2-form auxiliary field, v ab , a scalar auxiliary field D , agravitino ψ iµ and an auxiliary Majorana spinor χ i . Each vector mutiplet contains a 1-formgauge field A I , a scalar auxiliary field X I and a gaugino Ω Ii (where I = 1 , · · · , n v countthe number of vector multiplets) and i = 1 , SU (2) doublet index and α = 1 , · · · , r refers to U SP (2 r ) group. The hyper multiplet contains the auxiliary scalar fields A iα anda hyperino ζ α . We will use units G = π/ δψ iµ = D µ ε i + v ab γ µab ε i − γ µ η i ,δχ i = Dε i − γ c γ ab ˆ D a v bc ε i + γ ab ˆ R ab ( V ) ij ε i − γ a ε i ǫ abcde v bc v de + 4 γ ab v ab η i ,δ Ω Ii = − γ ab F Iab ε i − γ a ∂ a X I ε i − X I η i ,δζ α = γ a ∂ a A αi − γ ab v ab ε i A αi + 3 A αi η i , (2.5)where δ ≡ ¯ ǫ i Q i + ¯ η i S i + ξ aK K a and the covariant derivatives are defined by D µ ε i = (cid:18) ∂ µ + 14 ω abµ γ ab + 12 b µ (cid:19) − V iµ j ε j , (2.6)ˆ D µ v ab = ( D µ − b µ ) v ab = ∂ µ v ab + 2 ω c [ a v b ] c − b µ v ab , (2.7)in which b µ is a real boson in the Weyl multiplet and is SU (2) singlet [22].There is a well-known gauge to fix the conformal invariance of the off-shell formalismand reduce the superconformal symmetry to the standard symmetries of five dimensional N = 2 supergravity, A = − , b µ = 0 , V ijµ = 0 . (2.8)In this gauge the last equation of (2.5) gives η i in terms of ε i as, η i = 13 γ ab v ab ε i . (2.9)In the gauge (2.8) and using above equation (2.9) the supersymmetry variations (2.5)simplify to δψ iµ = (cid:0) D µ + v ab γ µab − γ µ γ ab v ab (cid:1) ε i ,δχ i = (cid:0) D − γ c γ ab D a v bc − γ a ǫ abcde v bc v de + ( γ ab v ab ) (cid:1) ε i ,δ Ω Ii = (cid:0) − γ ab F Iab − γ a ∂ a X I − X I γ ab v ab (cid:1) ε i . (2.10)In the next subsection we review the supersymmetric black ring solution of N = 2 fivedimensional supergravity in the presence of higher derivative supersymmetric corrections(2.3). To compute the entropy of an extremal black hole we just need to know the near horizongeometry. In [25] the near horizon of five dimensional supersymmetric black ring in thepresence of higher derivative terms (2.3) is derived using the entropy function formalism[33]. In addition by using the entropy function formalism the macroscopic entropy of black Here γ a a ··· a m = m ! γ [ a γ a · · · γ a m ] which is antisymmetric in all indices. Also the covariant curvatureˆ R ijµν is defined by ˆ R ijµν = 2 ∂ [ µ V ijν ] − V i [ µ k V kjν ] + fermionic terms , where V ijµ is a boson in the Weyl multipletwhich is in of the SU (2). For the solution we are going to consider, this term vanishes. Q i is the generator of N = 2 supersymmetry, S i is the generator of conformal supersymmetry and K a are special conformal boost generators of superconformal algebra [22]. ds = l AdS (cid:18) − r dt + dr r (cid:19) + l S ( dψ + e r dt ) + l S (cid:0) dθ + sin θdφ (cid:1) ,A I = e I rdt − p I θdφ + a I ( e r dt + dψ ) , X I = p I l AdS , D = 12 l AdS , (2.11) Q I = − C IJK p J a K , e I + e a I = 0 , v θφ = 38 l AdS sin θ, in which θ and φ are the coordinates of a usual 2-sphere and ψ is the coordinate of ring andis periodic, ψ ∼ ψ + 4 π , Q I are the electric charges and the radii are given by the magneticcharges p I , l AdS = l S = e l S = 12 (cid:18) C IJK p I p J p K + 112 c I p I (cid:19) / . (2.12)From (2.11) we can see that the metric consists of a U (1) fibred over an AdS base times atwo-sphere. In other words the isometries of the metric are SL (2 , R ) × U (1) × SO (3) andare generated by L = r∂ r − t∂ t , L = ( t + r − ) ∂ t − rt∂ r − e r ∂ ψ , L − = ∂ t , ¯ L = − e ∂ ψ , (2.13) J = − i∂ φ , J ± = e ± iφ ( − i∂ θ ± cot θ∂ φ ) . In fact we can think of the first part of the metric as locally
AdS with the symmetries SL (2 , R ) × SL (2 , R ) with the one of the SL (2 , R )’s broken to a U (1). One can show thatwe also have the locally defined killing vectors¯ L = e ψe (cid:18) r ∂ t + r∂ r − e ∂ ψ (cid:19) , ¯ L − = e − ψe (cid:18) r ∂ t − r∂ r − e ∂ ψ (cid:19) . (2.14)Notice that since ψ is periodic, ¯ L ± are not well defined globally. One finds that togetherwith the generator of the U (1) part of the near horizon isometry, ¯ L , we obtain an SL (2 , R )algebra, [ ¯ L m , ¯ L n ] = ( m − n ) ¯ L m + n , m, n = 0 , ± . (2.15)The periodicity of ψ breaks this SL (2 , R ) to a U (1). This local AdS symmetry will permitus to use the c-extremization approach [28] to find the associated central charge in section4. Now, the parameter e gives the angular momentum of the black ring solution in 5D,while if one reduces along the ψ direction, from a 4D point of view, e is an electric field. Inthe entropy function formalism one can not easily find this electric field in terms of physicalcharges (or angular velocity) of the black ring [25, 8].5olving the near horizon equations of motion leads to an additional relation betweenthe parameter e , magnetic charges p I of the black ring, electric charges Q I of the blackring and angular velocities J φ and J ψ , J φ − J ψ + 18 C IJ ( Q I − C IK p K )( Q J − C JL p L ) = 1 e (cid:18) C IJK p I p J p K + 16 c I p I (cid:19) . (2.16) J φ = 12 p I (cid:18) Q I − C IJK p J p K (cid:19) , (2.17)where C IJ is the inverse of C IJ ≡ C IJK p K . In [34], it was also shown that the left handside of eq.(2.16) is equal to the charge ˆ q , associated with the Kaluza-Klein photon, whichwill be used to compute the microscopic entropy of black ring in the next section,ˆ q ≡ J φ − J ψ + 18 C IJ ( Q I − C IK p K )( Q J − C JL p L ) . (2.18)The macroscopic entropy of supersymmetric five dimensional black ring is given by S mac = 2 πe (cid:18) C IJK p I p J p K + 16 c I p I (cid:19) = 2 π r ˆ q ( C IJK p I p J p K + c I p I )6 . (2.19)In the next two sections we will compute the microscopic entropy of black rings with higherderivative corrections (2.3). We will show that both formalisms lead to the same result forthe microscopic entropy and the macroscopic entropy calculated in this section (2.19). The microscopic entropy of extremal black rings can calculated by using the Kerr/CFTapproach. This approach can be applied when the near horizon geometry contains a U (1)fibred over AdS which is the case for black rings we consider.The Kerr/CFT approach was extended to the case with a Chern-Simons term [16].It was shown that for a theory with gravity and also other fields, the central charge isnot affected by non-gravitational fields. This approach was also generalized to theorieswith higher derivative corrections [17]. Although this generalization was based on fourdimensional kerr black hole in the extremal limit we will show that the black ring satisfythe conditions that help us to use the results of [17] to compute the central charge ofassociated Virasoro algebra in the presence of higher derivative corrections (2.3). Since the black ring near horizon geometry with higher derivative corrections is similar tothe case without, one can use the same boundary conditions as those used in [8], h µν ∼ O r /r /r r /r /r /r /r /r /r /r /r , (3.1)6n the basis ( t, r, θ, φ, ψ ). The generators associated to these boundary conditions are givenby ζ n = − e − inψ ∂ ψ − in r e − inψ ∂ r , (3.2)which satisfy a Virasoro algebra i [ ζ m , ζ n ] = ( m − n ) ζ m + n . (3.3)Two interesting facts can be noted when comparing (2.13) and (3.2). Firstly, ζ is propor-tional to ¯ L which is the generator of the near horizon U (1) symmetry. It is said that theVirasoro is “based” on this U (1). Secondly, the other ζ ’s do not commute with L whichis a generator of the near horizon SL (2 , R ). Furthermore, this non-commutativity is dueto the last term of L which is related to the fibration of the U (1) on an AdS base. Thismeans that the Virasoro is not decoupled from the SL (2 , R ).To apply the Kerr/CFT approach when higher derivative corrections are added it isuseful to do the calculations in a non-basis coordinates. The vielbeins associated to nearhorizon geometry of black ring are e ˆ t = l AdS rdt, e ˆ r = l AdS r dr, e ˆ θ = l AdS dθ, e ˆ φ = l AdS sin θdφ, e ˆ ψ = l S ( dψ + e r dt ) , (3.4)and the variations of the veilbeins are given by L ζ n e ˆ t = i n e − inψ e ˆ t , L ζ n e ˆ r = − e n e − inψ (cid:16) e ˆ ψ − e ˆ t (cid:17) , L ζ n e ˆ θ = L ζ n e ˆ φ = 0 , L ζ n e ˆ ψ = i n e − inψ (cid:16) e ˆ ψ − e ˆ t (cid:17) . (3.5)These variations are similar to the case of the Kerr black hole [17].The Virasoro algebra we found (3.3) corresponds to Poisson brackets between the gen-erators. Since we are interested in studying the quantum behavior of the boundary fluc-tuations, we need to find the Dirac brackets which may lead to a Virasoro algebra with acentral charge. To compute this central charge we follow [35, 36, 37]. The central charge isgiven by c ( k ) = 12 i Z ∂ Σ k invζ n [ L ζ − n g ; g ] (cid:12)(cid:12)(cid:12)(cid:12) n (3.6)where | n stands for the term of order n and k invζ n [ L ζ − n g ; g ] = − h X cd L ζ n ∇ c ζ d − n + ( L ζ n X ) cd ∇ [ c ζ d ] − n + L ζ n W c ζ c − n i − E [ L ζ n g, L ζ n g ; g ] , (3.7)in which covariant derivatives are defined with respect to the original metric g . X and W are related to Z abcd , the variation of the Lagrangian with respect to the Riemann tensor R abcd , Z abcd = δ cov LδR abcd , (3.8)by, ( W c ) c c c = − ∇ d Z abcd ǫ abc c c = 2( ∇ d X cd ) c c c . (3.9)7he process of finding the central charge for the supersymmetric black ring follows thesame recipe as for the Kerr solution. After some work one finds that the central chargeassociated to the Virasoro algebra (3.3) is, c ( k ) = − e Z Σ Z abcd ǫ ab ǫ cd vol(Σ) = 6 e π S mac . (3.10)In the last step we used the Iyer-Wald formula for macroscopic entropy of a black holewhich is generalization of Bekenstein-Hawking formula when the higher derivative termsare appeared. So the cental charge is c ( k ) = C IJK p I p J p K + c I p I . (3.11)As we shall see in the next section, this central charge is equal to the left central charge com-puted by the c-extremization formalism. This equality was shown for black rings withouthigher derivative corrections in [8]. Finding this relation for the case with higher derivativeterms is a much stronger result and unlikely to be a coincidence. We consider this equalityfurther in the discussion section. The microscopic entropy of supersymmetric black ring in the Kerr/CFT approach can becomputed by the following form of the Cardy formula, S ( k )mic = π c ( k ) T F T , (3.12)where T F T is the Frolov-Thorne temperature. The Frolov-Thorne temperature is an intrinsicfeature of metric and its definition is not corrected by higher derivative terms. So as usual,one can find the Frolov-Thorne temperature from the tψ cross term of near horizon geometry(3.4) T FT = 1 πe . (3.13)Using (3.12,3.13) one finds that S ( k )mic = 2 πe (cid:18) C IJK p I p J p K + 16 c I p I (cid:19) . (3.14)As we expect this microscopic entropy associated with the asymptotic symmetry groupis equal to the macroscopic entropy (2.19). One can also use the usual Cardy formula to compute the microscopic entropy of thesupersymmetric black ring. The low energy decoupling limit implies that only the left-handed excitations survive and the microscopic entropy is given by, S mic = 2 π r c L ˆ q . (4.1) Appendix B of [17] is devoted to this subject. Often there is a factor of 2 in the denominator of the expression for the Frolov-Thorne temperaturebut not in our case since we have take the period of ψ to be 4 π .
8n this form of the Cardy formula c L is the left central charge and ˆ q , given in eq.(2.18),corresponds to the left-handed excitations of the CFT . Using the c-extremization formalismone can compute this central charge from near horizon data. Although this formalism wasintroduced for a geometry with a globally AdS part, we assume that it can also be usedfor geometries which are locally AdS . We don’t prove this but the self-dual orbifold AdS perspective [26, 27] and EVH/CFT proposal [38] suggest that this approach can also be usedfor a locally AdS geometry. A posteriori the fact that the we obtain non-trivial agreementwith the results of the previous section is gives further weight to our assumption. We willdiscuss this point in last section.At the leading order it was shown that the c-extremization and Brown-Henneaux (orKerr/CFT) approach lead to the same result for the central charge [8]. At this level theleft and right central charges are equal and given by c L = c R = C IJK p I p J p K . (4.2)Turning on the higher order correction one can use the c-extremization approach to find theaverage of left and right central charges. Then, finding the gravitational anomaly gives thedifference between left and right central charges so that combining the two one can obtain c L and c R .The first step in applying the c-extremization formalism is choosing an appropriateansatz, ds = l AdS ds AdS + l S ds S , (4.3) A I = e I rdt − p I θdφ + a I ( e r dt + dψ ) , (4.4)Then by extremizing the c-function, c = 6 l AdS l S ( L + L ) , (4.5)with respect to, l AdS and l S , the AdS and sphere radii respectively, we find their values interms of the magnetic charges. The value of c-function at these radii gives the average ofleft and right central charges. Performing these calculation one finds, l AdS = 2 l S = (cid:18) C IJK p I p J p K + 112 c I p I (cid:19) / , (4.6) A I = e I rdt − p I θdφ + a I ( e r dt + dψ ) , X I = p I l AdS , D = 12 l AdS , (4.7) Q I = − C IJK p J a K , e I + e a I = 0 , v θφ = 38 l AdS sin θ, and the value of c-function at this extremum point is given by c | ext. = 12 ( c L + c R ) = C IJK p I p J p K + 34 c I p I . (4.8)There is a precise agreement between the above solution and the results of entropy functionformalism reviewed in the previous section (2.11).9n [28] it was shown that for the associated dual CFT the gravitational anomaly yieldsthe difference between left and right central charges, c L − c R = 12 c I p I . (4.9)Thus the left and right central charges are given by c L = C IJK p I p J p K + c I p I , c R = C IJK p I p J p K + 12 c I p I , (4.10)Now we can use the Cardy formula (4.1) and equations (2.18) and (2.16) to computethe microscopic entropy of black ring, S ( c )mic = 2 π r c L ˆ q πe (cid:18) C IJK p I p J p K + 16 c I p I (cid:19) . (4.11)The above entropy is in precise agreement with result of Kerr/CFT approach (3.14). In this paper we study the microscopic interpretation of SUSY black ring solutions of N = 2supergravity in the presence of supersymmetric completion of mixed gauge-gravitationalChern-Simons term (2.3). Because of the near horizon geometry of these solutions, one canuse both c-extremization and the Kerr/CFT approach to find the microscopic entropy viacomputing the associated central charge. We showed that central charge, which counts thedegeneracy of ground states in the CFT side, is given by the magnetic charges of SUSYblack rings (4.10) or (3.11) in both methods independently.We found that the usual form of the Cardy formula (4.1) without any subleading cor-rections can be used for SUSY black ring solution even in the presence of higher derivativecorrections (2.3). This works because the effect of the higher derivative corrections is rela-tively simple – essentially we just have a shift of the Kaluza-Klein photon charge (2.16,2.18).As long as we remain in the large charge regime, we do not need to consider subleadingcorrections. It means our results are in agreement with the canonical ensemble descriptionof black ring used in [40]. This discussion also applies to the use of (3.12).The most interesting result of our study is that the Kerr/CFT and c-extremizationapproaches which are apparently related to different Virasoro algebras lead to the samevalue of central charges and the same microscopic entropies in a highly non-trivial setting.In order compare these approaches, and to try get a handle on the various central chargesappearing in the game, it is helpful to consider the geometry in detail. While global
AdS has two SL (2 , R ) symmetries, as discussed in section 2, our near horizon only has one withthe other SL (2 , R ) is broken to a U (1). Now, it would seem to be natural to associate c L with the unbroken SL (2 , R ) and conclude that the right-handed excitations are killed by thebroken SL (2 , R ). This however seems to be incompatible with the fact that it is preciselythe residual part of the broken SL (2 , R ), ¯ L which forms the basis of the Virasoro algebraassociated with the asymptotic symmetry group considered in the Kerr/CFT approach. This was previously shown for BTZ black hole solutions of 3D gravity [39]. SL (2 , R ), L , L ± , do not commute with the Virasorogenerators of the Kerr/CFT approach (3.2) which means they are not independent. Thissuggests that although the asymptotic symmetry group Virasoro algebra is “based” on the U (1) part of the near horizon isometry, the SL (2 , R ) plays a crucial rule in Kerr/CFTcorrespondence.This leads us to conclude that both points of view are somehow talking to each other.The central charge counts the ground states dual to SUSY black rings and the number ofthese states is independent of any approach used to this counting.This is in agreement with the DLCQ approach which relates the left-handed excitationsof the self-dual orbifold AdS geometry on two distinct boundaries [41, 42, 43]. In ourcase, The S part, which carries the angular momentum of the black ring, is fibred overthe AdS part such that locally one has an AdS geometry called a self-dual orbifold ofAdS [26, 27]. From this viewpoint it was shown that extremal BTZ black hole, which hasself-dual orbifold AdS near horizon geometry, is dual to discrete light cone quantised CFT which admits one chiral Virasoro algebra [41, 42, 43].Our results suggest some further possible avenues to investigate. It would be interestingto compare three dimensional extremal BTZ black holes in the presence of some higherderivative terms and light cone quantised CFT . Another interesting avenue to consider iswhether hidden conformal symmetries appear beyond the extremal limit for SUSY blackrings with higher derivative corrections. In this situation one expects both left and rightcentral charges to be excited. Acknowledgements
We would like to thank M. M. Sheikh-Jabbari for many helpful comments and insights ona preliminary draft of this work and Robert de Mello Koch and Vishnu Jejjala for usefuldiscussion. This work is based upon research supported by National Research Foundation.Any opinion, findings and conclusions or recommendations expressed in this material arethose of the authors and therefore the NRF do not accept any liability with regard thereto.
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