aa r X i v : . [ h e p - t h ] J u l Preprint typeset in JHEP style - HYPER VERSION
Black holes in higher spin supergravity
Shouvik Datta, Justin R. David
Centre for High Energy Physics, Indian Institute of Science,C.V. Raman Avenue, Bangalore 560012, India. shouvik, [email protected]
Abstract:
We study black hole solutions in Chern-Simons higher spin supergravitybased on the superalgebra sl (3 | U (1) gauge fieldand a spin 2 hair in addition to the spin 3 hair. These additional fields correspondto the R-symmetry charges of the supergroup sl (3 | N = 2 super- W symmetry, we identify the bulk charges and chemical potentials with those of theboundary CFT. From these identifications we see that a suitable set of variables tostudy this black hole is in terms of the charges present in three decoupled bosonicsub-algebras of the N = 2 super- W algebra. The entropy and the partition functionof these R-charged black holes are then evaluated in terms of the charges of the bulktheory as well as in terms of its chemical potentials. We then compute the partitionfunction in the dual CFT and find exact agreement with the bulk partition function. ontents
1. Introduction 12. Constructing higher spin black holes 4 sl (3 |
2) 52.3 Black holes in higher spin supergravity 7
3. Bulk equations of motion and Ward identities 8
4. The higher spin black hole 12
5. Black hole thermodynamics 15 λ ] 20
6. Conclusions 21A. OPEs of N = 2 super- W
1. Introduction
The study of higher spin theories ( s ≥
2) in Anti-de Sitter spaces have been the focusof many recent works (see [1, 2] for comprehensive reviews). These theories offer toymodels of AdS/CFT with enhanced symmetry and without the complications of theinfinite tower of massive string excitations. Higher spin theories in 3 dimensional– 1 –nti-de Sitter space are easier to formulate in terms of Chern-Simons theory whichrenders them more tractable [4, 5, 6, 7]. In three dimensions it is also possible toconsistently truncate to a finite set of higher spin fields. There are explicit proposalsfor the CFT duals of Vasiliev like theories in terms of minimal models for cases withand without supersymmetry [3, 8].Classical solutions in higher spin theories in 3 dimensional Anti-de Sitter spaceare easy to construct (see [9] for a recent review). This is because in terms ofthe Chern-Simons formulation they correspond to a flat connection. Black holesand conical defect solutions constructed in these theories have been used to studythe spectrum, nature of singularities with enhanced higher spin symmetry and theholographic renormalization group [10, 11, 12]. Recently the conditions under whicha classical solution is supersymmetric in a Chern-Simons higher spin theory wasprovided in [13, 14]. Supersymmetric classical solutions should correspond to chiralprimaries in the boundary CFT and a recent check of this fact has been done in [15].Conformal field theory in 2 dimensions with enhanced supersymmetry usuallyadmits redefinitions of the currents so that the bosonic sub-algebras mutually com-mute. Consider the simple case of a CFT with N = 2 supersymmetry. The bosonicpart of this algebra consists of the stress tensor T and the U (1) current J . Redefiningthe stress tensor as ˆ T = T − c J (1.1)ensures that the new Virasoro algebra commutes with the U (1) current. This re-definition has an important consequence for a BTZ black hole carrying U (1) charge.A bulk theory with N = 2 asymptotic algebra is a Chern-Simons theory based onthe supergroup sl (2 | U (1) gauge field. Consider the BTZ black hole along withthe U (1) gauge field. The metric of the black hole is unaffected by the U (1) gaugefield. Thus the entropy is independent of the presence of U (1) field. However in theAdS/CFT correspondence, it is − ˆ T which is identified with the stress tensor of thebulk [16, 17, 18]. This shift is due to the extra energy carried by the gauge field.The Cardy-like formula for the entropy now is given by S = 2 π s c (cid:18) L − c J (cid:19) (1.2)where L and J are the zero modes of the stress tensor and the U (1) current ofthe CFT. We wish to study this phenomena in the presence of higher spin fields.A natural and consistent framework to introduce an additional U (1) in Vasilievtheories is to study supersymmetric Chern-Simons theories based on the supergroup sl ( N | N − U (1) as a R-symmetry. The second See [19] for a derivation. – 2 –oal in this paper is to reproduce the partition function of black holes in higher spinsupersymmetric theories from the CFT.Our set up is the Chern-Simons theory based on the supergroup sl (3 |
2) whosebosonic field content include the graviton and a spin 3 field. There is an additionalspin 2 field and a U (1) gauge field corresponding to the R-symmetry part of thesupergroup sl (3 | U (1) field we use the correspondence of the bulk equationsof motion to the Ward identities of the CFT [20]. In this case the Ward identitiesarise from the semi-classical operator product expansions of the the N = 2 super- W which is the asymptotic algebra of the sl (3 |
2) theory . We see that indeed the stresstensor is shifted precisely as expected. However, there is also a shift of the spin 3charge which is proportional to the product of the spin 2 R-charge and the U (1)charge. The precise shifts are provided in Table 1. The bosonic part of the N = 2super- W algebra can be written into 3 mutually commuting algebras given by N = 2 super- W ⊃ W + ⊕ W − ⊕ u (1) (1.3)where W + is a Virasoro algebra and W − is the bosonic W algebra [28]. From theseshifts found in table 1, we see that a natural linear combination of the bulk charges aredirectly identified with the charges in the mutually commuting bosonic algebras. Wethen construct a black hole solution carrying these charges and evaluate its partitionfunction and entropy. This is done using the integrability conditions satisfied bythe holonomy equations. The partition function and the entropy are written bothin terms of the charges and the corresponding chemical potentials. We see that theblack hole entropy can be written in terms of contributions from the decoupled CFTs.The contribution from the W + part is due to a Virasoro algebra and the entropyfrom the CFT can be understood in terms of the Cardy formula. The contributionfrom the W − part to the partition function with higher spin charge has been recentlycomputed in [33]. Appealing to this result we show that the entropy of the blackhole in the higher spin super-Chern-Simons theory can be exactly reproduced formthe CFT. We then discuss the implications of these observations for black holes intheories dual to the Kazama-Suzuki models of [8].The organization of the paper is as follows: Section 2 reviews the generalitiesof higher spin super Chern-Simons theory and then details the sl (3 |
2) algebra. Wethen write down the general form of the Chern-Simons connection which we will beinterested in. In Section 3 the correspondence of the bulk equations of motion withthe semi-classical Ward identities of the N = 2 super- W algebra is used to obtainthe relationship between the bulk charges and chemical potentials with that of theboundary theory. From these we see that a natural linear combination of bulk chargesare directly identified with the charges of the decoupled CFTs given in (1.3). Section Asymptotic algebras of higher spin super Chern-Simons theories were studied in [6, 13]. – 3 – discusses the construction of the higher spin black hole with spin 3 field carryingthe spin 2 and U (1) R-charges. It then demonstrates the integrability conditionsare satisfied by the holonomy conditions. This implies the existence of consistentthermodynamic description of the black hole. We then evaluate the entropy and thepartition function of the black hole solution both in terms of charges and the chemicalpotentials. We then show that the partition function can be reproduced exactly fromthe CFT. The conditions for which the black hole is supersymmetric is also analyzedin this section. We then discuss the implications of these observations for black holesin supersymmetric higher spin theories based on the supergroup shs [ λ ]. Section6 contains the conclusions. Appendix A contains the details of the semiclassicalOPEs of the N = 2 super W algebra. Appendix B demonstrates the equivalenceof the entropy written in terms of charges to that written in terms of the chemicalpotentials. Note added:
While this manuscript was in preparation we received [21] whichhas some overlap with this paper.
2. Constructing higher spin black holes
It has been well-known that gravity and its supersymmetric extensions in 3-dimensionscan be described by a Chern-Simons theories [22, 23, 24, 25]. Black hole in such grav-ity theories have also been studied [26]. Higher spin gravity with spins 2 , , · · · N isdescribed by generalizing the gauge group from SL (2 , R ) × SL (2 , R ) to SL ( N, R ) × SL ( N, R ) [4, 1, 27]. This can also be phrased in terms of the higher spin algebra hs [ λ ] with λ = − N . These hs [ λ ] theories have gained considerable interest in thecontext of its interpretation in terms of holographic minimal models [2]. As in thecase for pure gravity one may also look for supersymmetric generalizations of higherspin gravity. The gauge group in such a case is given by shs [ λ ] [8]. For the caseof N = 2 higher spin supergravity the gauge group is given by sl ( N | N − G is givenby S = k π Z (cid:20) str (cid:18)
A dA + 23 A (cid:19) − str (cid:18) ¯ A d ¯ A + 23 ¯ A (cid:19)(cid:21) (2.1)where A = A aµ T a dx µ , with T a being the generators of the supergroup. The gaugeconnections are given in terms of the tetrad and vielbein of the background metricas A = ω + e ¯ A = ω − e (2.2)The equation of motion for the action above is the flatness conditions on the gaugeconnections and is given by dA + A ∧ A = 0 d ¯ A + ¯ A ∧ ¯ A = 0 (2.3)– 4 –he metric can be obtained from the gauge connection using g µν = 1 ǫ ( N | N − str( e µ e ν ) (2.4)Here ǫ ( N | N − is a normalization constant given by N ( N − . sl (3 | sl (3 |
2) and has N = 2 supersymmetry. This is thesemiclassical version ( c → ∞ ) of the global part of the N = 2 super- W algebra[28]. This is a refection of the fact that the asymptotic algebra for this case is thatof super- W and the dual superconformal field theory has this symmetry.This algebra has bosonic generators J , L , V and W corresponding to spins 1, 2, 2and 3 respectively. The algebra formed by just L is the usual SL (2 , R ) algebra. Thereare fermionic generators G ± and U ± corresponding to spins 3/2 and 5/2 respectively.We shall list explicit commutation relations for this algebra below.[ J, J ] = 0 , [ L m , L n ] = ( m − n ) L m + n , (2.5)[ V m , V n ] = ( m − n )( L m + n + κV m + n ) , [ W m , W n ] = ( m − n )(2 m + 2 n − mn − L m + n + κ V m + n ) , [ J, L n ] = 0 , [ J, V n ] = 0 , [ J, W n ] = 0 , [ L m , V n ] = ( m − n ) V m + n , [ L m , W n ] = (2 m − n ) W m + n , [ V m , W n ] = κ (2 m − n ) W m + n . where κ = ± (5 / i . Here the subscripts m, n on the generators L run from − , , W run from − , − , , ,
2. The commutationrelations between bosonic and fermionic generators are given by[ L m , G ± r ] = ( m − r ) G ± m + n , [ J, G ± r ] = ± G ± r , (2.6)[ L m , U ± r ] = ( m − r ) U ± m + r , [ J, U ± r ] = ± U ± r , [ V m , G ± r ] = ± U ± r + m , [ G ± r , W m ] = (2 r − m ) U ± r + m , [ V m , U + r ] = κ ( m − r ) U + m + r + (3 m − mr + r − ) G + m + r , [ V m , U − r ] = − κ ∗ ( m − r ) U − m + r − (3 m − mr + r − ) G − m + r , [ U + r , W m ] = κ (2 r − rm + m − ) U + r + m + (4 r − r m + 2 rm − m − r + m ) G + r + m , [ U − r , W m ] = κ ∗ (2 r − rm + m − ) U − r + m + (4 r − r m + 2 rm − m − r + m ) G − r + m . Here the subscripts r, s on G ± run from − /, / U ± run from − / , − / , / , /
2. Finally the anti-commutation rules between– 5 –he fermionic generators are given by { G ± r , G ∓ s } = 2 L r + s ± ( r − s ) J, { G ± r , G ± s } = 0 , (2.7) { G ± r , U ∓ s } = 2 W r + s ± (3 r − s ) V r + s , { G ± r , U ± s } = 0 , { U + r , U − s } = − κ ( r − s ) W r + s + (3 s − rs + 3 r − )( L r + s + κ V r + s )+ ( r − s )( r + s − ) J r + s , { U ± r , U ± s } = 0 . This algebra has 12 fermionic and bosonic generators each. The bosonic part of thesuperalgebra is given a direct sum of the subalgebras sl (3) ⊕ sl (2) ⊕ u (1). This can beexplicitly seen by defining new spin-2 generators, T ± in terms of L and V as follows T + m = −
13 ( L m + 2 iV m ) T − m = 13 (4 L m + 2 iV m ) . (2.8)Substitituting these redefintions in (2.5) we obtain[ T + m , T − n ] = 0 , [ T + m , W n ] = 0 , (2.9)The generators T + m obey the sl (2) algebra while the generators T − n , W m obey thecommutation relations of the sl (3) algebra[ T − m , T − n ] = ( m − n ) T − m + n , [ T − m , W n ] = (2 m − n ) W m + n , (2.10)[ W m , W n ] = 316 ( m − n )(2 m + 2 n − mn − T − m + n . The sl (3) algebra above is same as that of as the one given in equation (A.2) of[20] with σ = (3 / . For the sl (3) part we shall be using the same representationof the generators as given in [20] while the for the sl (2) part the representationin terms of Pauli matrices are used. For the u (1) generator the diagonal matrix( − , − , − , − , −
3) is used as mentioned in Section 61 of [29]. We will choosethe gravitational sl (2) generators to be that given by L m and the correspondingsupercharges to be G ± r which form the principal embedding of osp (2 |
1) in sl (3 | osp (2 |
1) in sl (3 |
2) in [34] and the renormalizationgroup flows between the various embeddings.
The decomposition of the super- W algebra The fact that the global generators of the bosonic part of super- W decomposes intoglobal sl (3) ⊕ sl (2) ⊕ u (1) also carries over to local generators. It is known that thebosonic subalgebra of the N = 2 super- W algebra also splits into three mutuallycommuting pieces as mentioned in [28]. The splitting can be written as N = 2 super- W ⊃ W + ⊕ W − ⊕ u (1) (2.11)– 6 – + contains just a spin-2 generator which is given by ( T + + c J ) in the large c limit. W − contains a spin-2 generator given by ( T − − c J ) and a spin-3 generatorwhich is given by ( W − c J V ) in the large c limit. The central charges for thesesubalgebras are c + = − c , c − = 43 c , c J = 1 (2.12)We have written the field redefinitions and the central charges in the large c limit.The full quantum version of this decomposition and the central charges in each ofthe commuting sectors are given in [28]. The W + is then a Virasoro algebra withcentral charge c + while W − is a W algebra with c − as its central charge. Thisdecomposition will play an important role when we evaluate the partition functionof the black hole from the dual CFT and show that it precisely agrees with thatobtained from the classical solution in the bulk. There has been many recent constructions of classical solutions (black holes andconical defects) in higher spin theories [20, 12, 10]. Similar constructions in higherspin supergravity were studied in [13, 14]. They turn out to be interesting in their ownright [11, 30, 31, 9] and also from their interpretation in the dual CFT [32, 33, 34].The thermodynamics of higher spin black holes turn out to be quite interestingand have been recently investigated in [35, 36, 37, 38, 39, 40]. Here we shall beinterested in constructing black hole solutions and studying their thermodynamicsin such higher spin supergravity theories. The trial gauge connections for the higherspin black hole embedded in sl (3 |
2) which we shall be considering here are as follows A = b − a ( x + ) b + b − db ¯ A = b − ¯ a ( x + ) b + b − db (2.13)where a = (cid:18) L − πk L L − − πk V V − + π kσ W W − + πk J J (cid:19) dx + + ( µW + w W + w W + w − W − + w − W − + ρV + v V + v − V − + ℓL − − γJ ) dx − (2.14)¯ a = − (cid:18) L − πk ¯ L L − − πk ¯ V V − + π kσ ¯ W W − − πk J J (cid:19) dx − − (¯ µW − + ¯ w W + ¯ w W + ¯ w − W − + ¯ w W + ¯ ρV + ¯ v V + ¯ v − V − +¯ ℓL + γJ (cid:1) dx + (2.15)One is motivated to choose deformations of asymptotically-AdS connections of thisform on basis of [7, 20]. The 14 undetermined functions {J , L , V , W , µ, ℓ, v − , v , ρ, w − , w − , w , w , w } – 7 –re allowed to depend on x + (= t + φ ) and x − (= t − φ ). It will be seen later that γ, ρ and µ are chemical potentials conjugate to J , V and W .
3. Bulk equations of motion and Ward identities
In this section we shall be taking a look at the constraints on J , L , V and W imposedby the flatness conditions. We then show that we will obtain the same constraintsusing the Ward identities from the operator product expansions of the N = 2 super- W algebra. In the process we will relate the charges and chemical potentials in thebulk to their counterparts in the boundary. This method was developed in bosonictheories in [20] and applied to supersymmetric theories in [41, 42]. The AdS/CFTdictionary for this case will be obtained. When these charges and currents are usedas thermodynamic variables to calculate the entropy and the partition function, theidentifications of these quantities with that of the super- W CFT will be importantto match the entropy and the partition function obtain from field theory calculations.
On substituting the connection (2.14) in the Chern-Simons equation of motion ( da + a ∧ a = 0) we get the following considering the coefficients of each of the generators w = − ∂ + µw = + 12 ∂ µ − πk ( L + i V ) µw − = − ∂ µ + 4 π k ∂ + ( L + i V ) µ + 10 π k ( L + i V ) ∂ + µw − = 124 ∂ µ − π k ∂ µ ( L + i V ) − π k ∂ + ( L + i V ) ∂ + µ − π k ∂ ( L + i V ) µ + 4 π k ( L + i V ) µ + iπ W ρ kσv = − ∂ + ρv − = 12 ∂ ρ + 3 πi kσ W µ − πk ( L + i V ) ρℓ = 3 πkσ W µ − πk V ρ (3.1) ρ used in the connection a is the chemical potential conjugate to V and should not be confusedwith the radial coordinate. – 8 – , V and W satisfy the following equations ∂ − J = kπ ∂ + γ (3.2) ∂ − L = − ∂ + W µ − W ∂ + µ + ∂ + V ρ + 2 V ∂ + ρ (3.3) ∂ − V = − k π ∂ ρ − i σ ∂ + W µ − i σ W ∂ + µ + ∂ + ( L + i V ) ρ + 2( L + i V ) ∂ + ρ (3.4) ∂ + W = 3 i W ∂ + ρ + i ∂ + W ρ + σk π ∂ µ − σ ∂ ( L + κ V ) µ − σ∂ ( L + κ V ) ∂ + µ − σ∂ + ( L + κ V ) ∂ µ − σ L + κ V ) ∂ µ + 64 πσ k ( L + κ V ) ∂ + ( L + κ V ) µ + 64 πσ k ( L + κ V ) ∂ + µ (3.5) In this section we shall try to obtain the same equations as Ward identities from theOPEs of the N = 2 super- W algebra [28]. The OPEs are listed in Appendix A. Spin-1 current, J The non-vanishing OPEs for operators along with J ( z ) are that of J ( z ) J ( w ) and J ( z ) W ( w ). The Ward identity is given by ∂ ¯ z J ( z, ¯ z ) = ∂ ¯ z Z d y ( J ( z ) J ( y ) γ ( y ) + J ( z ) W ( y ) µ ( y )) (3.6)here by J ( z, ¯ z ) we mean the expectation value h J i γ,ρ,µ , weighted with e π R ( γJ + ρV + µW ) .Upon using the OPEs and upon using ∂ ¯ z (cid:0) z (cid:1) = 2 πδ (2) ( z, ¯ z ) we get ∂ ¯ z J = − c ∂ z (cid:18) γ + 6 c V µ (cid:19) (3.7)On converting the above equation to the Lorentzian signature ∂ − → − ∂ ¯ z , ∂ + → ∂ z and using k π = c we can identify the bulk variables in (3.2) with those from thealgebra as J = J , γ bulk = γ + 6 c V µ . (3.8) Stress-tensor, T For this case the non-zero OPEs are that of T ( z ) J ( y ), T ( z ) V ( y ) and T ( z ) W ( y ). Thevariation is then given as ∂ ¯ z T ( z, ¯ z ) = ∂ ¯ z Z d y ( T ( z ) J ( y ) γ ( y ) + T ( z ) V ( y ) ρ ( y ) + T ( z ) W ( y ) µ ( y )) (3.9)Substituting the OPE we obtain ∂ ¯ z T = − (2 ∂ z W µ + 3
W ∂ z µ + 2 V ∂ z ρ + ∂ z V ρ + J ∂ z γ ) (3.10)– 9 –sing (3.6), converting to Lorentzian signature and comparing with (3.3) leads us to L = − (cid:18) T − c J (cid:19) , V = − V, W = 34 (cid:18) W − c J V (cid:19) ,ρ bulk = ρ + 6 c J µ, µ bulk = µ (3.11)along with the ones in (3.8) which remain consistent identifications. Thus, the Wardidentities of J and T are sufficient to give all the identifications between the bulk andalgebraic variables. We shall now find the Ward identities of V and W and comparethem with the constraints from the bulk equations of motion with the identificationsgiven in (3.8) and (3.11). This is performed as an additional consistency check ofthe identifications. The other spin-2 current, V The contributing OPEs in this case are that of V ( z ) V ( y ) and V ( z ) W ( y ). ∂ ¯ z V ( z, ¯ z ) = ∂ ¯ z Z d y ( V ( z ) V ( y ) ρ ( y ) + V ( z ) W ( y ) µ ( y )) (3.12)from which we get ∂ ¯ z V = − (cid:16) c ∂ z ρ + 2( T + κV − c J ) ∂ z ρ + ρ∂ z ( T + κV − c J )+ 2 c ( J ∂ z ( T + κV ) − ∂ z J ( T + κV )) µ + 3 ∂ z µC [3] + 2 ∂ z C [3] µ + ∂ z ( J µ ) (cid:19) (3.13)where C [3] is given in equation (A.10) in the appendix. The identifications in (3.8)and (3.11) can be consistently used in while comparing (3.4) with the Lorentzianversion of the above equation and using (3.6) and (3.10). The spin-3 charge, W We finally come to the case of the spin-3 charge, W . The contributions to thevariation arise from W ( z ) W ( y ), W ( z ) V ( y ) and W ( z ) J ( y ) OPEs ∂ ¯ z W ( z, ¯ z ) = ∂ ¯ z Z d y ( W ( z ) W ( y ) µ ( y ) ρ ( y ) + W ( z ) V ( y ) ρ ( y ) + W ( z ) J ( y ) γ ( y ))(3.14)Substituting the OPEs we get the following ∂ ¯ z W ( z, ¯ z ) = − c W ∂ z µ − B [2] ∂ z µ − ∂ z B [2] ∂ z µ − (2 B [4] + 9 ∂ z B [2] ) ∂ z µ − ( ∂ z B [4] + ∂ z B [2] ) µ − C [1] ∂ z ρ − C [3] − ∂ z C [2] ) ∂ z ρ − ( C [4] + ∂ z C [3] ) ρ + 2 V ∂ z γ (3.15)– 10 –ubstituting the B [ ] s and C [ ] s above from the equation (A.10) and upon using (3.6),(3.10), (3.13) we can see that this equation exactly matches with (3.5) with the iden-tifications given in (3.8) and (3.11). In doing this we require to make an additionalidentification c W = π k . This is consistent with the relation c W = c found in [28]which was arrived at using the Jacobi identities satisfied by the super- W algebra.Although the calculations for this specific Ward identity are quite involved, one canexplicitly see the matching by comparing each of the coefficients of the derivativesof the chemical potentials γ , ρ and µ . The identifications of the charges and chemical potentials in the boundary and thebulk are summarized in the following table
Spin Bulk Boundary1 J J Charges 2
L − ( T − c J )2 V − V W ( W − c J V )Chemical 1 γ γ + c V µ potentials 2 ρ ρ + c J µ µ µ Table 1 : Relating the charges and currents of the bulk with that of the CFT These identifications constitute the AdS/CFT dictionary for this higher spin blackhole background. It is interesting to note that the charges and chemical potentialsshift due to the presence of the R-symmetry part. Note that these identificationssatisfied several non-trivial checks. The structure of the N = 2 super- W OPE’swas used to obtain the identifications. The OPE’s are of-course constrained bysupersymmetry. This is an independent reason for the dual CFT to have the N = 2super- W symmetry.Motivated by the definition (2.8) for the decoupled generators, let us now definethe following linear combinations of the charges in the bulk. T + = − ( L + 2 i V ) , T − = ( L + i V ) , W − = W (3.16)the identifications for the charges become T + → T + + c J , T − → T − − c J , W − → W − c J V (3.17)Now the bulk charges T ± , W − are identified precisely with the combinations of theboundary currents for which the bosonic part of the N = 2 super- W algebra de-coupled into 3 bosonic sub-algebras as mentioned in subsection 2.2. This shows the Note that in our conventions which is the same as in [20], − ( T − c J ) is positive – 11 –atural identification of the decoupled operators in the CFT is in terms of the ‘decou-pled charges’ in the bulk. The combination of the boundary currents which resultsin the three decoupled bosonic algebra can also be thought of as cosetting out the U (1) . The reason is that, now currents T + + c J and T − − c J and W − c J V areuncharged with respect to the U (1). Let us again emphasize that this decouplingresulted due to the tight structure of the N = 2 super- W OPE’s. This phenomenonof decoupling is a property of the N = 2 super- W algebra and is present in thesupersymmetric minimal models dual to Chern-Simons theories based on the infinitedimensional supergroup based on shs [ λ ] as discussed in section 5.4.
4. The higher spin black hole
In this section we write down the connections of the higher spin black hole embeddedin the sl (3 |
2) theory. If all the charges and chemical potentials are assumed to beindependent of x ± , then from the equations of motion (3.1) we get the followingsolution a = (cid:18) L − πk L L − − πk V V − + π kσ W W − + πk J J (cid:19) dx + + (cid:18) µW − πk ( L + i V ) µW + ( π k ( L + i V ) µ + iπ kσ W ρ ) W − + ρV +( πi kσ W µ − πk ( L + i V ) ρ ) V − + ( πkσ W µ − πk V ρ ) L − − πk J J (cid:17) dx − (4.1)¯ a = − (cid:18) L − − πk ¯ L L − πk ¯ V V + π kσ ¯ W W − πk J J (cid:19) dx − − (cid:18) ¯ µW − − πk ( ¯ L + i ¯ V ) µW + ( π k ( ¯ L + i ¯ V ) µ + iπ kσ ¯ W ¯ ρ ) W + ¯ ρV − + ( πi kσ W µ − − πk ( ¯ L + i ¯ V ) ¯ ρ ) V +( πkσ ¯ W ¯ µ − πk ¯ V ¯ ρ ) L + πk J J (cid:17) dx + (4.2)The above connection reduces to that of the charged BTZ black hole embedded inthe gravitational sl (2) for which the connections are given solely in terms of L and J generators. This can be seen by setting µ = ¯ µ = W = ¯ W = ρ = ¯ ρ = V = ¯ V = 0.Note that this black hole is however not continuously connected to the higher spinblack hole of [20]. The reason is that even though if one sets ρ = ¯ ρ = V = ¯ V = 0,the coefficient of V − does not vanish. This point was mentioned to us by Matthias Gaberdiel. – 12 – .2 Black hole holonomy and integrability
We shall now investigate the holonomy for the higher spin black hole constructed.The holonomy is a gauge invariant and meaningful quantity in Chern-Simons theory.We shall also see that this observable will play a role in determining the entropy forthe black hole. We shall be determining the holonomies along the thermal circle andthen demand their eigenvalues to be the same as the one for the BTZ black hole.The holonomy around the Euclidean time circle ( z, ¯ z ) → ( z + 2 πτ, ¯ z + 2 π ¯ τ ) is H = b − e ω b , ¯ H = be ¯ ω b − (4.3)where ω = 2 π ( τ a + − ¯ τ a − ) , ¯ ω = 2 π ( τ ¯ a + − ¯ τ ¯ a − ) (4.4)For the gauge connections embedded in the sl (3 |
2) theory, ω has the block diagonalform sl (3) ⊕ sl (2). The eigenvalues of the holonomy matrix for the BTZ black holeis ( − πi, , πi ) and ( − πi, πi ).On defining α = ¯ τ γ, α = ¯ τ ρ, α = ¯ τ µ (4.5)the sl (2) part of the holonomy for the gauge connection given in (4.1) is ω sl (2) = − π ( α k + π J τ ) k π ( L +2 i V )( τ − iα ) k π ( τ − iα ) − π ( α k + π J τ ) k ! (4.6)On finding the eigenvalues of the above matrix and equating them ( iπ, − iπ ) we get α k + π J τ = 0 (4.7)Then (4.6) becomes ω sl (2) = π ( τ − iα )( L +2 i V ) k π ( τ − iα ) 0 ! (4.8)while the sl (3) part is given by ω sl (3) = π ( L + i V ) √− σα k π ( ( L + i V ) σ ( τ − iα ) +9 W α ) kσ π (cid:16) π ( L + i V ) σα − k W ( τ − iα ) (cid:17) k √− σ π (cid:0) τ − iα (cid:1) − π ( L + i V ) √− σα k π ( ( L + i V ) σ ( τ − iα ) +9 W α ) kσ − π √− σα π (cid:0) τ − iα (cid:1) π ( L + i V ) √− σα k (4.9) There exist two possibilities here regarding whether to equate an eigenvalue to iπ or − iπ .However, the condition on J and α we write down is common to both the cases. – 13 –he combinations appearing above motivates us to define a new set of variables asfollows η + = τ − iα , T + = − ( L + 2 i V ) , k + = − kη − = τ − i α , T − = ( L + i V ) , k − = k (4.10) W − = W The combinations for T ± and W − were mentioned previously in (3.16). This is dueto the fact that the bosonic part of sl (3 |
2) splits as sl (3) ⊕ sl (2) ⊕ u (1).We shall now demand that the eigenvalues of holonomy should equal that of theBTZ black hole. An equivalent way of saying this istr( ω sl (2) ) = − π , (4.11)tr( ω sl (3) ) = − π , (4.12)det( ω sl (3) ) = 0 . (4.13)The first condition gives η + = ik + p πk + T + (4.14)the second condition for α = 0 = W − gives η − | α =0= W − = ik − p πk − T − (4.15)for non-zero α and W we have256 π σα T − − πk − η − T − − πk − η − α W − − k − = 0 (4.16)and from the determinant condition we get2048 π σ α T − + 576 πσk − η − α T − + 864 πσk − α η − W − T − + 864 πσk − α W − − k − η − W − = 0 (4.17)These are the same conditions as (5.14) of [20] with L → T − , τ → η − and α → α .The integrability conditions are then given by ∂ W − ∂η − = ∂ T − ∂α (4.18) We now analyse the supersymmetry of this higher spin black hole. In [14] it wasshown that the Killing spinor can be written in terms of products of backgroundholonomies and odd-roots of the superalgebra. For calculational simplicity we shallrestrict ourselves to the case W = 0. – 14 –he matrix a φ is given in terms of the Cartan matrices H i of the superalgebra[29] as Sa φ S − = − πiµ T − k + i ( ρ + 2 i ) r π T − k ! H + πiµ T − k + i ( ρ + 2 i ) r π T − k ! H + (2 ρ + i ) 2 π T + k H ¯4 + (cid:18) π J k + γ (cid:19) J (4.19)where L , are defined in (4.10). Using the subalgebra for sl (3 |
2) one can derive thesupersymmetric conditions using λ r α ri ± (cid:18) π J k + γ (cid:19) = in i n i ∈ Z (4.20)where r is the index for the Cartan matrices and i is the fermionic direction whichone chooses. α ri s are the odd-roots of the superalgebra. The commutation relationswith the Cartan matrices with the fermionic generators are[ H , E ¯ ι, ] = E ¯ ι, , [ H , E ¯ ι, ] = − E ¯ ι, , [ H , E ¯ ι, ] = 0[ H , E ¯ ι, ] = 0 , [ H , E ¯ ι, ] = E ¯ ι, , [ H , E ¯ ι, ] = − E ¯ ι, (4.21)[ H ¯4 , E ¯4 i ] = − E ¯4 i , [ H ¯4 , E ¯5 i ] = − E ¯5 i The supersymmetric condition(s) for the fermionic direction ¯ ι = ¯4 , j = 2 using (4.20)then turn out to be − − πiµ T − k + i ( ρ + 2 i ) r π T − k ! + πiµ T − k + i ( ρ + 2 i ) r π T − k ! − (2 ρ + i ) 2 π T + k + (cid:18) π J k + γ (cid:19) = in (4.22)One can similarly find 5 more conditions for the other fermionic directions.
5. Black hole thermodynamics
The integrability conditions (4.18) can be used as Maxwell relations for studyingthermodynamics of black holes in higher spin supergravity. We are thus led to definethe conjugate variable relations as follows η − = i π ∂S sl (3) ∂ T − , α = i π ∂S sl (3) ∂ W − (5.1)and also η + = i π ∂S sl (2) ∂ T + (5.2)– 15 –he total entropy from the connection A is given by S = S sl (2) ( T + ) + S sl (3) ( T − , W − ) (5.3)This can be understood as follows. The gauge connection we are considering has ablock diagonal form. The Chern-Simons action can then be written as a sum of twocontributions from each of these blocks. Since the partition function involves e − I ,the entropy can be written as the sum above.Using (4.14) in (5.2) we get S sl (2) = 2 π p πk + T + (5.4)The entropy contribution from the sl (3) part can calculated as follows. We substitute(5.1) in the trace condition (4.16) and then solve the resulting differential equation.On basis of (4.15) and [20] one can choose the following anzatz S sl (3) = 2 π p πk − T − f ( y ) where, y = − k − W − σπ T − (5.5)Substituting this in the differential equation we get36 y (2 − y )( f ′ ) + f − f ( y ) = cos
16 arctan p y (2 − y )1 − y !! (5.7)Thus the final result for the entropy is S =2 π (cid:18)p πk + T + + p πk − T − f (cid:18) − k − W − σπ T − (cid:19) + q πk + ¯ T + + q πk − ¯ T − f (cid:18) − k − ¯ W − σπ ¯ T − (cid:19)(cid:19) (5.8)This has the series expansion S =2 π p πk + T + + 2 π p πk − T − (cid:18) k − πσ W − T − − k − π σ W − T − + · · · (cid:19) + barred part (5.9)Note that setting W − = V = 0, the entropy reduces to S = 2 π √ πk L . (5.10)Here we have used the definitions in (4.10) to rewrite the expression in terms of k and L . Thus we have obtained the expected answer of the entropy of the BTZ black hole– 16 –mbedded in the gravitational sl (2) (constructed out of L m (= T + m + T − m ) generators).This serves as an additional check of the fact that the the entropy is the sum of thecontributions from the sl (2) and the sl (3) part of the Chern-Simons action.Recently there has been a discussion of entropy of higher spin black holes us-ing the canonical definition of energy in [39, 40] called the ‘canonical formalism’.This method differs from that used by [20, 12] which relies on the existence of thepartition function and the compatibility with the first law called the ‘holomorphicformalism’. The difference arises due to choice of boundary terms. As mentionedin [40], the method [20, 12] is more suited from the CFT point of view due to theexistence of the partition function. The CFT computations agree precisely with thatof the holomorphic formalism [33]. Our goal in this paper is also to explain the bulkpartition function using the CFT. Therefore we have generalized the holomorphicformalism of [20] to obtain the entropy of the supersymmetric black hole. We wish to write the formula we just obtained for the entropy of the black hole (5.8)in terms of the chemical potentials. This shall enable us to match the answer fromconformal field theory. The partition function of the higher spin black hole embeddedin sl (3 | ⊕ sl (3 |
2) is Z gravity = D e πi ( α J + η + T + + η − T − + α W − ) E BH (5.11)This can be written in terms of the conventional charges and chemical potentials as Z gravity = D e πi (cid:16) τ L + α J − α V + 43 α W (cid:17) E BH (5.12)here ‘BH’ indicates that the quantity is evaluated for the higher spin black holebackground. We thus have the following conjugate variable relations α = i π ∂S∂ J , ∂ log Z∂α = 4 π i J (5.13) η + = i π ∂S∂ T + , ∂ log Z∂η + = 4 π i T + (5.14) η − = i π ∂S∂ T − , ∂ log Z∂η − = 4 π i T − (5.15) α = i π ∂S∂ W , ∂ log Z∂α = 4 π i W (5.16)In [32] it was shown that the partition function for the hs[ λ ] black hole couldbe written in terms of the chemical potentials. The approach taken was to assumea power series solution for L , W and other higher spin charges, plug them into We are considering just one sl (3 |
2) or the holomorphic part. – 17 –he holonomy conditions and solve for the unknown coefficients in the power seriesexpansions. Finally upon integrating the solutions for the charges, one can obtainthe partition function. On performing a similar analysis as the above we obtain (forthe holomorphic part with terms up to O ( α )) T + = − c + πη (5.17) T − = − c − πη − − c − σ πη − α − c − σ πη − α + · · · (5.18) W − = c − σ πη − α + 40 c − σ πη − α + · · · (5.19)The holonomy condition imposed the following relation on J and α . α k + π J τ = 0 (5.20)from which we get α = − τ J c , J = − cα τ (5.21)To study the thermodynamics, we have the following relations τ = i π (cid:18) ∂S∂ L (cid:19) L , V , W , α = i π (cid:18) ∂S∂ J (cid:19) J , V , W . (5.22)Using it in the second equation of (5.21) we get J = − c (cid:18) ∂S∂ J (cid:19) L , V , W (cid:18) ∂S∂ L (cid:19) − J , V , W = c (cid:18) ∂ L ∂ J (cid:19) S, V , W = 0 (5.23)This implies that the u (1) charge is forced to be zero by the holonomy condition andit does not contribute to the partition function. This conclusion can also be reachedfrom the second relation in (5.22). Since, the entropy is independent of J we obtain α = 0. From the holonomy conditions this then implies J = 0. However it wasimportant to keep track of the J to obtain the redefinitions given in Table 1.On integrating (5.17), (5.18) and (5.19) we obtain the partition function to belog Z gravity ( η + , η − , α ) = πic + η + + πic − η − (cid:18) σ α η − + 160 σ α η − + · · · (cid:19) (5.24)where we have used k ± = c ± /
6. On exponentiating the above, we get Z gravity = q − c/ (cid:18) πic − η − (cid:18) σ α η − − σ α η − + · · · (cid:19)(cid:19) (5.25)Upon using S = log Z ( η + , η − , α ) − π i ( η + T + + η − T − + α W ), the entropy is S ( η + , η − , α ) = iπc + η + + iπc − η − (cid:18) σ α η − + 160 σ α η − + · · · (cid:19) (5.26)– 18 –he equivalence of the above formula with the one in terms of the charges (5.9) isshown in Appendix B.Using (4.10) the partition function (5.26) reads as (with σ = 9 / Z gravity = πic τ (cid:18) α τ + α τ + · · · (cid:19) (5.27) We shall now calculate the partition function from the dual CFT. As was notedearlier the bosonic part of N = 2 super- W algebra splits as W + ⊕ W − ⊕ u (1). Theexpression for the partition function is then given by Z CFT = tr H " e πi (cid:18) ˆ η + ( T +0 + 12 c J − c +
24 )+ˆ η − ( T − − c J − c −
24 )+ˆ α ( W − c ( JV ) )+ˆ α J (cid:19) (5.28)Note that this is the natural CFT partition function one would write down sincethe N = 2 super- W algebra splits as W + ⊕ W − ⊕ u (1). The partition functionsimply evaluates the number of states weighted with the charges of decoupled bosoniccurrents. The chemical potentials appearing here are those in the CFT and are notthe same as those in gravity. We would now like to write the CFT partition functionin terms of the chemical potentials appearing the bulk. The relation between thetwo sets of chemical potential is given in Table 1. Using the relations in Table 1 andsimilar definitions as given in (4.5) and (4.10) for the chemical potentials in the CFTwe can obtain the the following identityˆ η + ( T +0 + c J ) + ˆ η − ( T − − c J ) + ˆ α J = η + ( T +0 + c J ) + η − ( T − − c J ) + α J (5.29)This identity allows the rewriting of the CFT partition function in terms of thechemical potentials of the bulk. Note that this non-trivial identity was arrived atby using the relations given in Table 1 which resulted from the structure of the N = 2 super W algebra. Here again supersymmetry played a role in this crucialsimplification. The partition function then becomes Z CFT = q − c/ tr H (cid:20) e πi (cid:16) η + ( T +0 + 12 c J )+ η − ( T − − c J )+ α ( W − c ( JV ) )+ α J (cid:17) (cid:21) (5.30)Since the W + , W − and u (1) are mutually commuting subalgebras, we have Z CFT = q − c/ tr H (cid:20) e πiη + ( T +0 + 12 c J ) e πi (cid:16) η − ( T − − c J )+ α ( W − c ( JV ) ) (cid:17) e πiα J (cid:21) (5.31)and also the full Hilbert space of states in the CFT factorizes as H = H + ⊕ H − ⊕ H u (1) (5.32)– 19 –he u (1) part does not contribute to the partition function for our case since weare restricting ourselves to the zero charge sector as seen in (5.23). The partitionfunction can then be written as contributions from the Virasoro ( W + ) and the W ( W − ) parts. Z CFT = q − c/ tr H + (cid:20) e πiη + ( T +0 + 12 c J ) (cid:21) tr H − (cid:20) e πi (cid:16) η − ( T − − c J )+ α ( W − c J V ) (cid:17) (cid:21) (5.33)We shall calculate the partition function at high temperature regime 1 /τ →
0. Theleading term for the Virasoro part is 1. Plugging in the answer calculated for the W part from [33] (with λ = 3, c → c − , τ → η − and α → α ) , we get Z CFT ≃ q − c/ (cid:18) πic − η − (cid:18) σ α η − − σ α η − + · · · (cid:19)(cid:19) (5.34)The above answer for the partition function matches precisely with the one calculatedfrom gravity (5.25). It can also be seen easily that (5.34) reduces to the Cardy’sformula for the BTZ black hole embedded in the gravitational sl (2) when α = 0 = α = α . λ ] One can generalize the above observations found for black holes in the sl (3 |
2) the-ory to the case of sl ( N | N −
1) theory and the Chern-Simons theory based on thesuperalgebra shs[ λ ]. The crucial observation which made it possible to obtain thepartition function of the black hole from the CFT was the existence of charges bywhich the CFT decoupled into three bosonic sub-algebras. The charges of the blackhole could then be mapped to charges in each of the sub-algebras. The black hole westudied turned out to have higher spin charge in the sl (3) sub-algebra as well as acharge in the sl (2) part. Thus appealing to the result of [33] and the Cardy formulathe entropy of the black hole was reproduced from the CFT.Let us now examine the situation for the Chern-Simons theory based on thesuperalgebra shs[ λ ]. It is dual to the Kazama-Suzuki model based on the coset [8] SU ( N + 1) k × SO (2 N ) SU ( N ) k +1 × U (1) N ( N +1)( k + N +1) (5.35)The model has a W N +1 superalgebra whose global part is sl ( N + 1 | N ). The t’ Hooftlimit of this coset with λ = Nk + N (5.36)is dual to the shs[ λ ] Chern-Simons theory. As discussed in [43, 8] the coset admitsthe following decomposition SU ( N + 1) k × SO (2 N ) SU ( N ) k +1 × U (1) N ( N +1)( k + N +1) ∼ SU ( k ) N × SU ( k ) SU ( k ) N +1 × SU ( N ) k × SU ( N ) SU ( N ) k +1 × U (1)(5.37) [32, 33] has σ set to − – 20 –hese cosets are precisely of the form considered by [2] in the minimal model/higherspin duality. Thus the bosonic part of the W N +1 superalgebra decomposes into thealgebra W ∞ (1 − λ ) ⊕ W ∞ ( λ ) ⊕ u (1). It’s worthwhile noting that the coset is invariantunder the level-rank exchange N ←→ k (or λ ←→ − λ ). These theories thereforehave a strong-weak self-duality. Now any black hole considered in the shs[ λ ] theorywill carry a specific set of bosonic charges. The relation (5.37) implies that, thereexists a re-definition of the charges such that they correspond to charges in thedecoupled bosonic sub-algebra of the superalgebra. Note that the decomposition in(5.37) is a property of these N = 2 supersymmetric minimal models, the structureof the super-algebra is important for this decomposition just as we have seen in ourstudy of the N = 2 super W algebra. The partition function of black holes carryinga specific set of higher spin charges constructed by [32] in the W ∞ ( λ ) theory has beenreproduced in the CFT by the computation done in [33]. Therefore we conclude thatthe partition function of higher spin black holes constructed by [32] embedded in W ∞ (1 − λ ) ⊕ W ∞ ( λ ) algebra will be reproduced in the CFT. It will be interesting tofind the re-definitions of the charges of the black hole so that they can be mapped tocharges in the decoupled algebras for the shs[ λ ] just as we have done for the sl (3 |
6. Conclusions
In this paper we have obtained the relations between the definitions of the chargesand the chemical potentials between the boundary CFT and the bulk sl (3 |
2) Chern-Simons theory. These relations are summarized in Table 1. From these relations weobserved that a natural linear combinations of the bulk charges can be identified withthe charges of the decoupled bosonic sub-algebras of the super- W CFT. We thenconstructed a higher spin black hole in this theory and evaluated both its entropyand partition function. The entropy was shown to be a sum of contributions from thedecoupled sub-algebras. The decomposition of the CFT into decoupled sectors alsoenabled the evaluation of the partition function which was shown to be in preciseagreement with that evaluated in the bulk.Let us now summarize where supersymmetry plays a role in our analysis. Firstlythe question of studying the effect of the U (1) charge in Vasiliev theories is naturallyrealizable in theories based on supergroup sl ( N | N −
1) which contains a U (1) aspart of the R-symmetry. Secondly the shifts in the definition of the charges andthe chemical potentials given in Table 1 was obtained from the detailed analysis the N = 2 super W -algebra. Finally the fact that this superalgebra decouples as in(2.11) and the redefinitions of Table 1 were crucial in the simplification of the CFTpartition function which is summarized in the identity (5.29). This identity relatedthe CFT chemical potentials to that of the bulk which enabled the evaluation and– 21 –he comparison of the partition functions evaluated in the boundary CFT and thatfrom the bulk.As we have discussed these observations can be generalized to the case of Chern-Simons theory based on the supergroup shs[ λ ]. It will be interesting to find theanalog of the relations given in Table 1 for this case. These relations will facilitatethe evaluation the entropy of the higher spin black hole in these theories. Anotherdirection to explore is the implication of the relations in Table 1 and its counterpartsfor smooth conical defects in these theories. It is possible that these classical solutionscan be related to the vacuum. Acknowledgements
We would like to thank Michael Ferlaino, Matthias Gaberdiel, Rajesh Gopakumar,S. Prem Kumar and Amitabh Virmani for useful discussions. S.D. thanks ETH,Zurich and ICTP, Trieste for hospitality during which a part of this work was com-pleted. The work J.R.D. is partially supported by the Ramanujan fellowship DST-SR/S2/RJN-59/2009.
A. OPEs of N = 2 super- W Following [28], we shall list the operator product expansions which were used tocalculate the Ward identities in Section 2. J ( z ) J ( w ) ∼ c/ z − y ) (A.1) J ( z ) W ( w ) ∼ z − w ) V ( w ) (A.2) T ( z ) J ( w ) ∼ J ( w )( z − w ) + ∂ w J ( w )( z − y ) (A.3) T ( z ) V ( w ) ∼ V ( w )( z − w ) + ∂ w V ( w )( z − y ) (A.4) T ( z ) W ( w ) ∼ W ( w )( z − w ) + ∂ w W ( w )( z − y ) (A.5)– 22 – ( z ) V ( w ) ∼ A [2] ( z − w ) + ∂ w A [2] ( z − w ) + c/ z − w ) (A.6) V ( z ) W ( w ) ∼ z − w ) C [4] + (cid:18) z − w ) + 1( z − w ) ∂ w (cid:19) C [3] + (cid:18) z − w ) + 5( z − w ) ∂ w + 1( z − w ) ∂ w (cid:19) C [2] + 6( z − w ) C [1] (A.6) W ( z ) J ( w ) ∼ z − w ) V ( w ) + 2( z − w ) ∂ w V ( w ) (A.7) W ( z ) V ( w ) ∼ z − w ) ( − C [2] + 6 ∂ w C [1] ) + 1( z − w ) (3 C [3] − ∂ w C [2] + 3 ∂ w C [1] )+ 1( z − w ) ( − C [4] + 2 ∂ w C [3] − ∂ w C [2] + ∂ w C [1] ) + 6( z − w ) C [1] (A.8) W ( z ) W ( w ) ∼ c W z − w ) + (cid:18) z − w ) + 1( z − w ) ∂ w (cid:19) B [4] + (cid:18) z − w ) + 30( z − w ) ∂ w + 9( z − w ) ∂ w + 2( z − w ) ∂ w (cid:19) B [2] (A.9)The exact expressions of A [ ] , B [ ] and C [ ] appearing in the OPEs above are givenin [28]. However, we are interested in the semiclassical limit of c → ∞ . It can seenfrom the equations of motion that the operators L , V and W scale as O ( c ), while allthe chemical potentials scale as O (1). On retaining just the O ( c ) terms in the OPEs A [ ] , B [ ] and C [ ] are expressed as follows A [2] = T − c J + κVB [2] = 14 (cid:16) T + κ V (cid:17) − c J B [4] = 24 c T − c J T + 92 c J ∂ J − c ∂ ( J ) + 24 ic T V + 9 ic J WC [1] = 12 JC [2] = 0 C [3] = 4 c J T − c J + i W + 7 ic J V (A.10) C [4] = 2 c ( J ∂ ( T + κV + 32 c J ) − ∂J ( T + κV + 32 c J ))While obtaining the above the fermionic operators were set to zero. The last term here had a 36 instead of 6 in [28] as a possible typographical error. The OPE wehave written here gives the correct commutation relation. – 23 – . Equivalence of the black hole entropy formulae
We shall prove the equivalence of the formula for the entropy of the higher spin blackhole written in terms of charges (5.8) or (5.9) with that of the formula in terms ofthe chemical potentials (5.26).The formula for the entropy in terms of the chemical potentials reads as S ( η + , η − , α ) = iπc + η + + iπc − η − (cid:18) σ α η − + 160 σ α η − + · · · (cid:19) (B.1)We assume a series expansion for S ( T + , T − , W − ) of the form S ( T + , T − , W − ) = 2 π p πk + T + + 2 π p πk − T − ∞ X j =0 χ j (cid:18) W − T − (cid:19) j (B.2)One can substitute (B.2) on the LHS of (B.1) and find α = i π ∂S∂ W and η i = i π ∂S∂ L i and substitute them on the RHS of (B.1), to get the following equations χ = 1256 πσχ = 3 kχ (B.3)5 (cid:0) k χ + 18 k χ + 524288 π σ χ (cid:1) = 3072 πkσχ (cid:0) χ + 4 χ (cid:1) ...which have the solutions χ = 1 , χ = 3 k − πσ , χ = − k − π σ , · · · (B.4)This is in precise agreement with the expansion in terms of the charges given in (5.9). References [1] M. Vasiliev,
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