An Entropy Formula for Higher Spin Black Holes via Conical Singularities
aa r X i v : . [ h e p - t h ] M a r IPMU13-0023YITP-13-4
An Entropy Formula for Higher Spin BlackHoles via Conical Singularities
Per Kraus a, and Tomonori Ugajin b,c, a Department of Physics and Astronomy,University of California, Los Angeles, CA 90095,USA, b Kavli Institute for the Physics and Mathematics of the Universe,University of Tokyo, Kashiwa, Chiba 277-8582, Japan c Yukawa Institute for Theoretical Physics, Kyoto University,Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan,
Abstract
We consider the entropy of higher spin black holes in 2+1 dimensionsusing the conical singularity approach. By introducing a conical singular-ity along a non contractible cycle and carefully evaluating its contributionto the Chern Simons action, we derive a simple expression for the entropyof a general stationary higher spin black hole. The resulting formula isshown to satisfy the first law of thermodynamics, and yields agreementwith previous results based on integrating the first law. e-mail: [email protected] e-mail: [email protected] ontents Higher spin gravity theories in anti de Sitter space [1, 2] have played an impor-tant role in exploring new versions of holography, e.g. [3, 4, 5, 6, 7, 8, 9, 10, 11].The theories themselves are interesting because they are expected to be toy mod-els of string theory in large curvature spacetimes. In 2+1 dimensions, higherspin fields are topological, as is the gravitational field, and so we can formulatethe theory in terms of Chern Simons theory [12, 13, 14].Recently, black holes with higher spin charges in 2+1 dimensional higherspin theories have been extensively studied [15, 16, 17, 18, 19, 20, 21, 22, 23, 24,25, 27, 34, 26, 28], and it turned out [16] that they do not have event horizons ingeneral since the metric in higher spin theories is gauge dependent. Nevertheless,there does exist a gauge invariant notion of the regularity of the (Euclidean)configuration, and one can assign thermodynamic properties to the black holes[15]. In [15] the black hole entropy is derived by demanding it satisfy the firstlaw of thermodynamics. Via the AdS/CFT correspondence, these black holesmap to generalized thermal ensembles of CFTs with higher spin symmetries,and where comparison is possible the black hole and CFT entropies are foundto agree [21, 26].Despite this progress, some aspects of higher spin black holes are still unclear.First of all, a general formula which calculates the entropy of all higher spin blackholes is not known. It is important to find an analog of the Wald formula [40] forthe higher spin theory. Second, there are in fact a few different approaches toderiving the entropy of higher spin black holes, and the assumptions and resultsdo not always agree [15, 22, 23, 28]. What is the relation between these differentapproaches? Finally, a gauge invariant understanding of the causal structure ofhigher spin black holes is unavailable in general [16, 20].One approach to black hole entropy in ordinary gravity is the conical deficitmethod [29, 30, 31, 32]. Recall that the entropy is obtained from the partition1unction as S ( β ) = − (cid:18) β ∂∂β − (cid:19) log Z ( β ) . (1)Applied to black holes there are two possible of this formula. In general, thepartition function is obtained from the action of the Euclidean black hole withtime periodicity β . In the first interpretation one considers a smooth Euclideanmetric at arbitrary β , evaluates the action, and then differentiates as above. Torespect smoothness, the parameters of the black hole such as the mass mustvary along with β . In the second interpretation, one keeps all parameters fixedwhile varying β . Changing β away from its preferred value thereby introduces aconical singularity at the horizon. Carefully evaluating the contribution of thissingularity to the action and evaluating (1) one obtains an alternative expressionfor the entropy. It turns out that these two approaches yield the same answer foran arbitrary diffeomorphism theory of gravity, and in particular they coincidewith the Wald entropy [29, 30, 31, 32].One advantage of the conical singularity approach is that it makes it manifestthat the entropy is associated with the local geometry at the horizon. Anotherinteresting aspect is that it is closely linked to methods used in computingentanglement entropy. When we compute the entanglement entropy S A of regionA in the time slice, we first introduce a deficit angle δ = 2 π (1 − n ) on ∂A whichis the boundary of the region A. Then we evaluate the partition function Z n of the theory on the singular manifold with the deficit angle. Entanglemententropy is derived by taking the derivative of Z n [41], S A = (cid:18) n ∂∂n − (cid:19) (cid:12)(cid:12)(cid:12) n =1 log Z n (2)Within the context of AdS/CFT the connection is even sharper, as eternal blackholes can be regarded as pure states in a tensor product of two CFTs, and theentropy arises after tracing over one copy [37, 38, 39].The concept of the regularity of a configuration in the 3d higher spin theoriesis correctly defined as triviality of the holonomy of the connection [15, 36]. Thisis a direct generalization of the regularity of the metric in the spin 2 case.Actually, at least for spin 3 black holes, the holonomy condition and the usualregularity condition are equivalent in the gauge where the metric of the blackhole has an event horizon [16].Then we expect that we can derive a Wald like formula by evaluating theaction of the higher spin black hole with a conical singularity. In this paper wegeneralize the conical singularity method to the Chern Simons gauge theory in2+1 dimensions. A sketch of the procedure is as follows: starting from a blackhole connection with trivial holonomy around the timelike cycle which satisfiesthe correct thermodynamical relation, we deform the period of the Euclideantime direction so that the holonomy around the cycle becomes slightly non-trivial. The non triviality of the holonomy indicates that the field strength ofthe corresponding connection has a delta function divergence along a non con-tractible cycle. We evaluate the Chern Simons action of the singular connection2y appropriately regularizing the connection, following the approach of [32] inthe metric formulation. We find the following general formula for the entropyof a stationary higher spin black hole, S = − πik Tr [ A + ( τ A + − ¯ τ A − )] − πik Tr (cid:2) A − (cid:0) τ A + − ¯ τ A − (cid:1)(cid:3) . (3)Here A and A denote the two Chern-Simons gauge fields. The spacetime coor-dinates are x ± and the radial coordinate ρ . The formula for S is independentof ρ . As far as we are aware, this is the first paper which generalizes the conicalsingularity approach to the connection formalism of gravity.To verify that this formula is physically sensible we show that it obeys thefollowing first law variation δS = − π i ∞ X s =2 α s δW s − π i ∞ X s =2 α s δW s , (4)where W s denote the spin-s charges, and α s their conjugate thermodynamicpotentials (and the same for the barred versions). Previously [15], the blackhole entropy was derived by assuming (4), and then integrating. Here we findthat the solution to this problem is given by (3). It follows immediately that(3) will reproduce previous results based on integrating the first law.This note is organized as follows. In section 2 we review the method toevaluate the Einstein-Hilbert action of a metric with conical singularity, andapply it to derive the entropy formula of black holes. In section 3 we adapt theprocedure to the connection formalism. In section 4 we demonstrate that theresulting entropy formula satisfies the correct first law variation for an arbitraryhigher spin black hole. This confirms that the entropy formula derived in thisway correctly reproduces all previous results based on integrating this first law.Appendix A contains some computations related to the general stationary blackhole. Note:
As this manuscript was being prepared the paper [33] appeared, whicharrives at our entropy formula by a different route and shows that it obeys thefirst law. This paper also greatly clarifies the relation between the differentapproaches to computing the entropy of higher spin black holes.
In this section, we review the derivation of the Wald formula by evaluating thegravitational action of a conical singularity. In subsection 2.1, we evaluate theEinstein-Hilbert action of a conical singularity in two dimensions by taking thelimit of a regularized metric. Since a Euclidean black hole with deficit angleon the bifurcation surface looks like a direct product of a two dimensional coneand the bifurcation surface near the tip, we can use the result of 2.1 to evaluatethe action of the singular black hole. We then discuss its relation to the Waldformula in subsection 2.2. 3 .1 Evaluation of the action of a conical singularity in 2dimension
In this section, we review the evaluation of the Einstein-Hilbert action of ametric with a conical singularity on a two dimensional manifold [29, 30, 31, 32].As an example, consider the metric: ds = e Φ( r ) ( dr + r dθ ) . (5)Let us assume θ ∼ θ + 2 πα, α = 1. The metric has a conical singularity atthe tip when the period of the θ direction is not 2 π . It is convenient to embedthe cone into R by the map: x = rα sin θα y = rα cos θα z = p − α r, (6)so the cone is mapped to the surface1 − α α ( x + y ) − z = 0 (7)with metric ds = e Φ( r ) ( dx + dy + dz ), the pull back of which gives the metric(5) on the 2d plane. In this form, the singularity is manifest because ∂z∂x , ∂z∂y areindeterminate at the tip of the cone z = 0.Since the metric (5) has a conical singularity, the curvature of the metriccontains a delta functional divergence at the tip in addition to the ordinaryregular part, R ∼ R reg + (1 − α ) δ ( r ). The coefficient in front of the deltafunction is attached so that for a regular α = 1 metric, only R reg appears in R .To see this, we would like to evaluate the contribution of the singularity tothe Einstein Hilbert action R √ gR . Two steps are needed. First we construct afamily of smooth metrics g µν ( a ), each labeled by a real positive number a , anddemand that they approach to the original singular one in the limit a →
0. Wecall the family the “regularization” of the singular metric. One way to constructa regularization is by modifying the surface (6) by a function f ( r, a ) as z = p − α f ( r, a ) , ∂ r f ( r, a ) | r =0 = 0 , lim r →∞ f ( r, a ) → r, (8)with the ambient metric ds = e Φ( r ) ( dx + dy + dz ) held fixed. We also assume f ( r,
0) = r . For example, if we take z = r a + 1 − α α r , (9)then the surface is replaced by a smooth hyperboloid when a = 0. For general f ( r, a ), the pull back of the metric on the 2D plane is modified as ds = e Φ( r ) ( u ( r ) dr + r dθ ) u ( r ) = α + (1 − α )( ∂ r f ( r, a )) . (10)4e can see these metrics are regular at the tip r = 0. Second, we evaluate theEinstein Hilbert actions I ( a ) of the regularized metrics for general non zero a .Since these metrics are regular, we can safely evaluate I ( a ), I ( a ) = 2 πα Z ∞ dr u ′ ( r ) u − Z ∞ dr Z πα rdθ p u ( r )∆Φ( r )= 4 π (1 − α ) − Z ∞ dr Z πα rdθ p u ( r )∆Φ( r ) , (11)where ∆ is the Laplacian of the metric without conformal factor (depending on u ( r )). The a → a → I ( a ) = 4 π (1 − α ) − Z ∞ dr Z πα rdθ ∆Φ( r ) . (12)Since the second term of the expression can be derived by directly substitutingthe metric (5) into the action, it describes the contribution from the regular partof the curvature to the action. However, there is an additional term. The firstterm in (12) is interpreted as the contribution of the conical singularity at thetip, because it vanishes when α = 1. The result show that the scalar curvatureof the metric (5) is given by √ gR = √ gR reg + 2(1 − α ) α δ ( r ) , (13)as we expected. It is important to note that the contribution of the singularityto the action does not depend on the regularization function f ( r, a ) we use. Thisassures us that the value 4 π (1 − α ) is intrinsic to the conical singularity. We can generalize the result to higher dimensions if the manifold we consider isthe direct product of a 2 dimensional cone C α = S α × R and a smooth manifoldΣ. In the case of the metric of the form ds = e Φ( r ) ( r dθ + dr ) + ds , (14)we are assuming θ ∼ θ + 2 πα, α = 1. Demanding that the volume of the S α × Σlocated at r = r is held fixed, the general regularization of the metric can bewritten ds = e Φ( r ) ( r dθ + u ( r ) dr ) + ds , u (0) = α , u ( ∞ ) = 1 (15)Note that only g rr is allowed to change. This turn out to be the correct reg-ularization and from this one can compute various geometric invariants in thepresence of the conical deficit. The results turn out to be equivalent to thestatement that the Riemann tensor contains a delta functional singularity on Σ, R µναβ = ( R reg ) µναβ + 2 π (1 − α )( n µα n νβ − n µβ n να ) δ Σ , (16)5here δ Σ is delta function on Σ which satisfie Z C α × Σ δ Σ √ g = Z Σ √ h, (17)and n µα = n µ n α + n µ n α , where n , n denotes the vector fields normal to Σ .Euclidean black holes with temperature different from the Hawking temper-ature are examples of these geometries. In this case Σ is the bifurcation surfaceof the black hole.Suppose Z ( β ) is the quantum gravity partition function with fixed temper-ature β , evaluated semiclassically as Z ( β ) = Z [ Dg ] e − I E [ β,g ] ≃ e − I Ec [ β,g c ( β )] δI E [ β, g ] δg (cid:12)(cid:12)(cid:12) g = g c ( β ) = 0 , (18)where I E denotes the Euclidean action. Then the entropy S ( β ) of the systemis given by S ( β ) = (cid:18) β ∂∂β − (cid:19) I E [ β, g c ( β )] . (19)Note that semiclassical metric depends on β because of regularity.There is an alternative way of computing S ( β ), S ( β ) = (cid:18) α ∂∂α − (cid:19) (cid:12)(cid:12)(cid:12) α =1 I Ec [ αβ, g c ( β )] . (20)This expression instructs to first evaluate the Euclidean action for the fixedmetric g c ( β ) but with varying time periodicity αβ . Such geometries with α = 1have conical singularities, and their action can be calculated by the methodreviewed in the previous section. By substituting (16) into (20) one immediatelyobtains the Wald formula [31, 32]. S = 4 π Z Σ √ h ∂ L ∂R µναβ n µα n νβ . (21)Note that if we divide the action into the regular part and the singular part, I Ec [ α, g c ( β )] = I Ereg + I Esing , the former does not contribute to the entropy, (cid:18) α ∂∂α − (cid:19) I Ereg = 0 , (22)because I Ereg is proportional α as we saw in (12). In this section we adapt the discussion of the previous section to the Chern-Simons formulation. We focus on BTZ black holes for our explicit computations,but then propose that the resulting entropy formula holds in general. Thevalidity of this proposal will be confirmed in the next section.6 .1 Entropy of the BTZ black hole
It is well known that 3d Einstein gravity with negative cosmological constant canbe formulated in terms of Chern Simons theory with the gauge group SL (2 , R ) × SL (2 , R ), I [ A, ¯ A ] = I CS [ A ] − I CS [ ¯ A ] , (23) I CS [ A ] = k π Z Tr (cid:18) A ∧ dA + 23 A ∧ A ∧ A (cid:19) , (24)with k = 1 / G . For definiteness, in this section we consider the 2 × L i , L j ] = ( i − j ) L i + j , Tr(L L − ) = − , Tr(L L ) = 12 . (25)The connections are related to the vielbein e and spin connection ω as A = ω + e, ¯ A = ω − e, (26)and the metric is given by g µν = 2Tr[e µ e ν ] . (27)The connections for the nonrotating BTZ black hole can be taken as A = e ρ + (cid:0) e r L − e − r L − (cid:1) dx + + L dr (28)¯ A = − e ρ + (cid:0) e r L − − e − r L (cid:1) dx − − L dr, (29)where the horizon is at r = 0 and e ρ + = q π L k , where L is proportional to themass.The metric of the black hole is ds = 4 e ρ + (cid:0) − sinh r dt + cosh r dθ (cid:1) + dr . (30)The value of the entropy of the black hole is obtained by the Bekenstein-Hawkingformula S = A G = 4 kπ β , (31)where β is the inverse Hawking temperature β = πe − ρ + .We now show how to derive this by the conical singularity method. If wekeep the connections fixed but identify the time coordinate as t ∼ = t + iαβ thenwe introduce a conical singularity in the metric for α = 1. This can be seen atthe level of the connections by evaluating the holonomies around the imaginarytime circle, e H A = (cid:18) cos πα − ie − r sin πα − ie r sin πα cos πα (cid:19) , (32) e H A = (cid:18) cos πα − ie r sin πα − ie − r sin πα cos πα (cid:19) . (33)7or a nonsingular metric we need the holonomies to be in the center of SL(2,R),which requires α to be an integer.Now we would like to evaluate I CS [ A ] for this connection. We proceedby regularizing the connection, evaluating its action, and then removing theregulator.It is convenient to use a rescaled Euclidean time coordinate T , t = iαβT , sothat the coordinate periodicity is fixed as T ∼ = T + 1 . In this coordinate thesingular connection is written A = iαβA t dT + A θ dθ + L dρ ¯ A = iαβ ¯ A t dT + ¯ A θ dθ − L dρ, (34)and the corresponding singular metric is g T T ( S ) = −
12 ( αβ ) Tr (cid:0) A t − ¯ A t (cid:1) (35) g θθ ( S ) = 12 Tr (cid:0) A θ − ¯ A θ (cid:1) . (36)There are various way to regularize the connection. For example, consider˜ A = A t u ( r ) ( iβαdT − cdθ ) + ( cA t + A θ ) dθ + L dρ ˜¯ A = ¯ A t u ( r ) ( iβαdT + cdθ ) + (cid:0) − c ¯ A t + ¯ A θ (cid:1) dθ − L dρ, (37)where c is some constant. The connections (37) are regular provided u (0) = α . We also demand u ( ∞ ) = 1 so that we go back to the original one at theboundary. A ρ and ¯ A ρ are unchanged because we are working on the gaugewhere A ρ = L , ¯ A ρ = − L .Below we fix the value of c that appears in the connections so that theregularization is consistent with the regularization of the metric (15). Themetric components g T T ( R ) and g θθ ( R ) of these regularized connection near thetip r ∼ g T T ( R ) = − β Tr (cid:0) A t − ¯ A t (cid:1) (38) g θθ ( R ) = 12 Tr (cid:20) c (cid:0) A t + ¯ A t (cid:1) (cid:18) − α (cid:19) + (cid:0) A θ − ¯ A θ (cid:1)(cid:21) . (39)From these expressions one notices how the metric components change by theregularization. In particular, in the BTZ case, δg T T = g T T ( R ) − g T T ( S ) = 2(1 − α ) g T T ( S ) , δg θθ = − c (1 − α ) g θθ ( S ) . (40)We used the property of the BTZ connection at r = 0, namely, A t + ¯ A t = A θ − ¯ A θ .As in the metric case (15), we demand that the volume of the torus located at r = r , r << c appearing in the regularization(37) has to be 1. 8ow that we have specified the regularization, we can compute the action I CS [ A ] of the singular configuration via regularization. As we are only interestedin terms which are proportional to (1 − α ), we only have to calculate the R Tr A ∧ dA term, since the A term is proportional to α and vanishes in (20) k π Z Tr A ∧ dA = k π Z Tr [( A t + A θ ) A t ] ∧ (cid:18) − u ′ ( r ) u ( r ) dr (cid:19) ∧ iαβdT ∧ dθ + I reg = − ikβ − α )Tr [( A t + A θ ) A t ] + I reg , (41)where I reg denotes the terms which are proportional to α . Similarly, k π Z Tr ¯ A ∧ d ¯ A = − ikβ − α )Tr (cid:2) ( − ¯ A t + ¯ A θ ) ¯ A t (cid:3) + ¯ I reg . (42)Since the Euclidean action is related to the Chern Simons actions via iI E = I CS [ A ] − I CS [ ¯ A ], we derive the expression for the entropy of the BTZ black holeby using (20), S = kβ A t + A θ ) A t ] + kβ (cid:2) ( ¯ A t − ¯ A θ ) ¯ A t (cid:3) (43)= 4 kπ β . The result reproduces the Bekenstein Hawking formula (31).This result can be generalized to the rotating BTZ black hole. In this casewe have both an inverse temperature β and the angular velocity of the horizonΩ. These can be combined to form τ = iβ π (1 + Ω) and τ = − iβ π (1 − Ω). Forthe Euclidean black hole, τ plays the role of the modular parameter of theboundary torus. Repeating the above analysis for this case we find the result(see Appendix A) S = − πik Tr [ A + ( τ A + − ¯ τ A − )] − πik Tr (cid:2) ¯ A − (cid:0) τ ¯ A + − ¯ τ ¯ A − (cid:1)(cid:3) , (44)which indeed yields the correct entropy of the rotating BTZ black hole.Although this result was derived for the BTZ black hole, since the formuladoes not make any specific reference to this solution we propose that it holdsmore generally. It is not obvious that this is a correct assumption. In particular,in the above argument we didn’t consider all possible regularizations of thesingular connection, and the argument for setting the constant c = 1 is notentirely compelling. Furthermore, the connection representing BTZ is not ofthe most general form. Fortunately, we can check that the result is correct byverifying that it obeys the correct first law variation. We carry this out in thenext section. In the preceding section we have motivated a simple expression for the entropyof a higher spin black hole. In this section we wish to verify that the result is9ndeed correct, and can be applied to general higher spin black holes. Our maintool is the first law of thermodynamics: in the thermodynamic limit the entropyis defined to be the object whose variation satisfies the first law, and so if wecan establish this property then we are done.To keep the discussion as general as possible, we consider a theory with aninfinite tower of higher spin charges, ( W , W , . . . ), where W s denotes a spin-scharge. Here we focus on just the “holomorphic” or ”leftmoving” charges, buteverything we say has an obvious parallel on the anti-holomorphic or rightmov-ing side. Each conserved charge has a corresponding conjugate potential, andwe denote these as ( α , α , . . . ). We will interchangeably use a different nota-tion for the spin-2 versions: W ↔ L and α ↔ τ , which are identified as theholomorphic stress tensor and modular parameter.Following [15] we think in terms of an underlying partition function of theform Z = Tr h e π i P ∞ s =2 α s W s i . (45)The right hand side has a precise meaning on the CFT side of the AdS/CFTcorrespondence, but here is just being used as a mnemonic for motivating theform of the first law. Namely, we have δS = − π i ∞ X s =2 α s δW s . (46)Next, let us recall the general rules for constructing higher spin black holes,and identifying their charges and potentials. We work in the context of hs[ λ ] × hs[ λ ] Chern-Simons theory, and recall that upon setting λ = ± N this theoryreduces to SL(N,R) × SL(N,R) Chern-Simons theory. In fact, it will become clearthat our derivation will apply to any Lie algebra with an SL(2,R) subalgebra,which includes hs[ λ ] as a special case.The Lie algebra hs[ λ ] has generators V sm , with s = 2 , , . . . and m = − ( s − , . . . s −
1. An SL(2,R) subalgebra is furnished by V ± , . The trace operationobeys Tr( V sm V tn ) ∝ δ s,t δ m, − n (47)and in particular we write Tr( V ss − V s − ( s − ) = t s . (48)Another useful fact is that V V ss − = V ss − V = V s +1 s .As is standard we write the connection as A = b − ab + b − db , b = e ρV (49)with a = a z dz + a z dz . (50)10ote that we are working in Euclidean signature. The component a z is takento be in highest weight gauge [8] a z = V + ∞ X s =2 c s W s V s − ( s − . (51)The constants c s are fixed by demanding that the charges W s obey the algebraof W ∞ [ λ ]. In particular, c is fixed to be c = 2 πt k . (52)Next we need to specify a z . To define a flat connection it has to commutewith a z , which can be satisfied by taking a z to depend on powers of a z , a z = ∞ X s =2 f s +1 ( a z ) s (cid:12)(cid:12)(cid:12) traceless (53)where f s are coefficients. The s = 1 term is absent, since it can be removedby redefining the coordinates ( z, z ). Noting the property ( V ) s = V s +1 s we canwrite a z = ∞ X s =2 f s +1 ( V s +1 s + . . . ) (54)where . . . denote generators with small value of the lower mode index. Restoringthe ρ dependence, the leading terms displayed above give the leading large ρ behavior.The coefficients f s are fixed by the holonomy conditions. The Euclideanblack hole has coordinates identified as ( z, z ) ∼ = ( z + 2 π, z + 2 π ) ∼ = ( z + 2 πτ, z +2 πτ ). Assuming constant a , the holonomy around the τ cycle is H = e ω , ω = 2 π ( τ a z + τ a z ) . (55)A smooth solution is obtained provided H lies in the center of the gauge group,which requires that ω has certain fixed eigenvalues. We can impose these con-ditions by requiring the Tr( ω n ), n = 2 , , . . . , take fixed values; for instance, forblack holes smoothly connected to BTZ, we demand that these traces coincidewith their BTZ values. These equations are in one-to-one correspondence to thefree parameters f s , and can be used to fix their values.The constants f s are proportional to the potentials α s appearing in the firstlaw. This was originally shown by a Ward identity analysis [15]. As will becomeclear momentarily, the relation is f s = − πkc s t s α s τ (56)11o that we have a z = − πk ∞ X s =2 α s c s t s τ ( V s +1 s + . . . ) . (57)The holonomy relations can now be used to express the potentials α s in termsof the charges W s , or vice versa.Written in terms of the holonomy, the candidate entropy formula is S = − ik Tr( a z ω ) (58)together with its anti-holomorphic partner. We now verify that this obeys thecorrect first law of thermodynamics. We have δS = − ik Tr( δa z ω ) − ik Tr( a z δω ) . (59)However, it’s easy to see that the second contribution vanishes. The conditionthat the traces of ω take fixed values implies that δω = [ ω, X ] for some X .Using this, along with [ a z , ω ] = 0, which follows from the fact that ω is builtout of powers of a z , we readily verify Tr( a z δω ) = 0. The variation of a z is δa z = ∞ X s =2 c s δW s V s − ( s − . (60)Inserting this and taking the trace yields δS = − π i ∞ X s =2 α s δW s . (61)as desired.Without doing any computations, we can establish that our entropy formulawill agree with the results obtained in [15, 17, 18, 19, 21]. This is becausethose computations were based on solving the holonomy conditions and thenintegrating the first law variation. Here we have shown that if the holonomyconditions are imposed then our entropy formula obeys the correct first lawvariation. Therefore, it must agree with previous results.Let us make some further comments. In the original work [15] it appearedsomewhat miraculous that that the first law variation could be consistentlyintegrated; this required verifying the integrability constraints, which turned outto follow rather non-transparently from the holonomy condition. Our discussionhere removes the mystery surrounding this procedure. In particular in (59) wesee very explicitly that if the holonomy is not kept fixed then δS will acquire anadditional unwanted term. This makes it clear that one should fix the holonomyin order to obtain the desired first law. Note also that any fixed holonomy willdo, in terms of satisfying the first law.Another notable point is that our derivation extends essentially automati-cally to to an arbitrary gauge group containing an SL(2,R) subgroup. Simply12ecompose the the generators into irreducible representations of SL(2,R), anddenote the generators in a given representation as V ( i ) sm , m = − ( s − , . . . s − i label takes into account the multiplicity of a given representation.Note though that in our discussion we took the index s to obey s >
2, whichleaves out the singlet; including the s = 1 case is straightforward. For somegauge groups, such as SL(N,R) with N >
2, there are multiple inequivalentchoices of SL(2,R) subgroups, and these lead to the existence of black holeswith different asymptotics [16]. From the CFT point of view, these correspondto thermal states in CFT with different W-algebra symmetries. Since nowherein our computation did we assume a particular SL(2,R) subgroup, it should beclear that our entropy formula applies to all such cases.
In this paper we obtained a formula that computes the entropy of a generalstationary higher spin black hole in 2+1 dimensions. Although the notion of anevent horizon is somewhat vague for black holes in higher spin theories, we dohave a definite notion of a conical singularity in terms of holonomy. A Euclideanblack hole with the period of the timelike cycle different from the inverse of theHawking temperature is an example of a configuration with a conical singularity.Then the field strength of the configuration is delta function divergent at thesingularity. It was recognized [29, 30, 31, 32] that only the contribution fromthe conical singularity to the action is necessary to reproduce the entropy of theblack hole.With this point of view, in this paper we developed a method to calculatethe contribution of the conical singularity to the Chern Simons action, by care-fully regularizing the connection. We used this result to compute the blackhole entropy. An advantage of this method is that since only terms with radialderivatives in the action have a chance to produce a delta function like diver-gence, it is sufficient to evaluate the R A ∧ dA term in the Chern Simons action.We don’t need to care about other terms, such as boundary terms, which arerequired when evaluating the total free energy of the black hole. More precisely,since these terms are all proportional to α (deficit parameter), they vanish in(20).This statement is equally true in the metric formulation of Einstein grav-ity. When we evaluate the free energy of the asymptotically flat Schwarzchildblack hole, the bulk Einstein-Hilbert action vanishes and we have to take intoaccount the Gibbons-Hawking boundary term. On the other hand, when wecompute the entropy of the black hole by the conical singularity method, weonly have to evaluate the contribution of the conical singularity at the tip tothe Einstein-Hilbert action. Since the entropy only depends on the local geom-etry of the horizon, it is efficient to use a computational scheme that makes thisfact manifest.It is useful to compare the general status of our entropy formula with that ofthe area law or Wald formula. A nice property of the latter is that they can be13valuated on any cross section of the horizon. Therefore, they make sense evenwhen applied to non-stationary black holes, such as those that are absorbinginfalling matter, although the physically correct formula for dynamical blackhole entropy might need to include additional terms in order to guarantee itsmonotonicity in time. On the other hand, our formula depends explicitly on theparameter τ , which depends on the black hole temperature. Since the notion oftemperature only makes sense in thermodynamic equilibrium, we don’t expectto be able to apply our result to out of equilibrium black holes. An outstandingchallenge is therefore to find an entropy formula which does make sense out ofequilibrium.Another feature that could be improved is that our formula only appliesto black holes and not thermal AdS. If one blindly plugs in the gauge con-nection for thermal AdS into our entropy formula one finds a nonzero result,whereas the correct answer is of course zero. The area law or Wald entropyautomatically assign zero entropy to thermal AdS, simply because there is noevent horizon. The reason why this is inconvenient is the following. Giventhe entropy, the partition function is obtained via Legendre transformation asln Z = S + 4 π i P s α s W s . For BTZ black holes a very useful observation isthat Euclidean BTZ and thermal AdS are related by a coordinate transforma-tion that acts as a modular transformation on the boundary. The partitionfunction is obtained from the Euclidean action and so is invariant under co-ordinate transformations, and this makes modular invariance of the partitionfunction manifest. Among other things, this provides a very convenient way ofcomputing the partition function and entropy of BTZ. This strategy fails whenapplied to our formalism, because our entropy formula, and hence the partitionfunction derived from it, only applies to black holes and not thermal AdS. Thuswe cannot use it to establish the modular properties of the partition function,which would be very useful in order to make direct contact with the CFT. Acknowledgements
We would like to thank T. Azeyanagi, J. de Boer, D. Fursaev, S. Hellerman,E. Perlmutter for discussions, and T.Takayanagi for reading the manuscriptcarefully, frequent discussions, encouragement and support. P.K. is supportedin part by NSF grant PHY-07-57702. T.U. is supported by World PremierInternational Research Center Initiative (WPI Initiative), MEXT, Japan, andby JSPS Research Fellowships for Young Scientists.
Appendix A. Entropy formula for general station-ary black holes
In this appendix we derive an entropy formula for general stationary higher spinblack holes. An example of a stationary black hole is a rotating BTZ black hole.14he connection of the black hole is given by A = (cid:18) e ρ L − e − ρ π L k L − (cid:19) dx + + L dρ (62)¯ A = − (cid:18) e ρ L − − e − ρ π ¯ L k L (cid:19) dx + − L dρ with L 6 = ¯ L . The event horizon is located at e ρ + = 2 πk p L ¯ L . (63)The entropy of the black hole is given by S = A G = 2 π (cid:16) √ π L k + p π ¯ L k (cid:17) . (64)In the corresponding Euclidean configuration, the modular parameter τ of theboundary torus contains non vanishing real part, τ = iβ π (1 + Ω) = ik √ πk L , (65)where Ω is the complex angular velocity. Since lines Θ = θ − Ω t = const arecontractible cycles, it is convenient to write the connection as A t dt + A θ dθ = iβ ( A t + Ω A θ ) dT + A θ d Θ= 2 π ( τ A + − ¯ τ A − ) dT + A θ d Θ . (66)We introduced a rescaled Euclidean time coordinate T which satisfies t = iβT and T ∼ = T + 1.The holonomy around the contractible cycle is derived by integrating theconnection along the line Θ = 0. When (65) is satisfied the holonomy lies inthe center of the gauge group. Suppose we replace β → αβ appearing in theconnection and vary α away from 1 while the relation T ∼ T + 1 is kept fixed.Then the connection develops a conical singularity and the holonomy becomesnontrivial. To evaluate the action, we have to regularize the connections. Itturns out that the connections˜ A = 1 u ( r ) (2 π ( τ A + − ¯ τ A − ) dT − A t d Θ) + ( A t + A θ ) d Θ˜¯ A = 1 u ( r ) (cid:0) π (cid:0) τ ¯ A + − ¯ τ ¯ A − (cid:1) dT + ¯ A t d Θ (cid:1) + (cid:0) − ¯ A t + ¯ A θ (cid:1) d Θ (67)are the right regularization because they do not change the volume of the toruslocated at ρ, ρ − ρ + << R A ∧ dA term and taking the derivative in terms of α , we get the final expression, S = − πik Tr [ A + ( τ A + − ¯ τ A − )] − πik Tr (cid:2) ¯ A − (cid:0) τ ¯ A + − ¯ τ ¯ A − (cid:1)(cid:3) . (68)It is straightforward to verify that this yields agreement with (64).15 eferences [1] M. A. Vasiliev, “Higher spin gauge theories: Star product and AdS space,”In *Shifman, M.A. (ed.): The many faces of the superworld* 533-610[hep-th/9910096].[2] X. Bekaert, S. Cnockaert, C. Iazeolla and M. A. Vasiliev, hep-th/0503128.[3] I. R. Klebanov and A. M. Polyakov, “AdS dual of the critical O(N) vectormodel,” Phys. Lett. B , 213 (2002) [hep-th/0210114].[4] E. 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