Partition functions of higher spin black holes and their CFT duals
aa r X i v : . [ h e p - t h ] S e p Partition functions of higher spin black holesand their CFT duals
Per Kraus and Eric Perlmutter Department of Physics and AstronomyUniversity of California, Los Angeles, CA 90095, USA
Abstract
We find black hole solutions of D = 3 higher-spin gravity in the hs[ λ ] ⊕ hs[ λ ] Chern-Simons formulation. These solutions have a spin-3 chemical potential, and carry nonzerovalues for an infinite number of charges of the asymptotic W ∞ [ λ ] symmetry. Applying apreviously developed set of rules for ensuring smooth solutions, we compute the black holepartition function perturbatively in the chemical potential. At λ = 0 , λ we expect that our gravity result will match the partitionfunction of the coset CFTs conjectured by Gaberdiel and Gopakumar to be dual to thesebulk theories.August 2011 [email protected], [email protected] . Introduction Higher spin theories of gravity, as developed by Vasiliev and collaborators, are fasci-nating theories that lie, in some sense, halfway between ordinary gravity and string theory;see [1,2] for reviews. In particular, while they share with string theories such features asinfinite towers of higher spin fields and nonlocal dynamics, their full (classical) equations ofmotion can be written down in a background independent manner. Relatively recently, ithas been realized that higher spin theories may lead to soluble examples of the AdS/CFTcorrespondence [3,4,5,6,7,8,9,10,11,12,13,14,15,16].Given the major role that black holes play in attempts to understand quantum gravityand holography, we hardly need to justify the motivation for constructing and studyingblack holes in higher spin theories. Here, extending previous work [17,18], we focus on theD=2+1 dimensional higher spin theory [19,20,21], and we will consider black holes thatgeneralize the BTZ solution [22]; see [23,24,25] for other work on black holes in higher spingravity.In ordinary gravity, the asymptotic symmetry algebra consists of left and right movingVirasoro algebras [26], and the general rotating BTZ black hole carries independent leftand right moving Virasoro zero mode charges. The BTZ entropy takes the form of Cardy’sformula, and so matches elegantly with the entropy of any CFT dual with the same centralcharge [27].In higher spin gravity, the asymptotic symmetry algebra contains one additional (leftand right moving) conserved charge for each higher spin field [14,28,29,30]. We expectthere to exist black holes that carry these conserved charges. If these black holes can befound and their entropy computed, we can use these results to test any proposed AdS/CFTduality involving these theories, assuming that the entropy can be computed on the CFTside as well.Apart from AdS/CFT applications, understanding how to compute the entropy ofhigher spin black holes represents an interesting technical and conceptual challenge: dueto the nonstandard form of these theories, one cannot apply (at least with current under-standing) such familiar approaches as the area law, the Wald entropy [31], or the Gibbons-Hawking Euclidean action [32]. This problem was studied in [17,18] in the simplest versionof higher spin gravity, based on SL(3,R) × SL(3,R) Chern-Simons theory. Black hole solu-tions were found and their entropy computed by appealing to first principles, in particularto the fact that what constitutes a physically satisfactory entropy is that it appear cor-rectly in a thermodynamical first law. The other novel ingredient, reviewed in more detailbelow, was to give a gauge invariant characterization of a smooth event horizon, based onthe holonomies of the Chern-Simons connection. The usual approach of determining theHawking temperature by compactifying imaginary time and demanding the absence of aconical singularity is not straightforward to apply in this context, as it is not fully gaugeinvariant. Indeed, as was shown explicitly in [18], even the existence of an event hori-zon in the metric is a gauge dependent statement: a gauge transformation was exhibitedthat transformed the metric between a black hole and a traversable wormhole. Despite all1hese subtleties, by following the logic in [17,18] it is possible to unambiguously computeall physical properties of these higher spin black holes.In this paper we apply our previous logic to the Vasiliev theories containing an infinitetower of higher spin fields [21]. These theories are based on a one-parameter family ofinfinite dimensional gauge algebras, denoted hs[ λ ] [33]. The BTZ black hole is still asolution of this theory, but rather too simple as it carries vanishing values for all higherspin charges. To access the higher spin sector we turn on a nonzero spin-3 chemicalpotential, α . Due to the nonlinear structure of the theory, this triggers nonzero valuesfor the entire infinite tower of higher spin charges. The values of these charges, and thefull smooth solution, can be determined systematically using perturbation theory in α . Asin the SL(3,R) case studied in [17,18], crucial input is provided by demanding a gaugeinvariant smooth horizon, as expressed in terms of the holonomies. The main output ofthis procedure is a result for the black hole partition function, Z ( τ, α ; τ , α ). Here τ is themodular parameter of the torus that describes the boundary of the Euclidean black holegeometry, and α is the leftmoving spin-3 chemical potential, as noted above. Similarly, α isthe rightmoving analog of α . As usual, given the partition function, other thermodynamicalquantities such as the energy and entropy can be obtained by suitable differentiation. Ourresult for the black hole partition function, up to order α , isln Z = iπk τ " − α τ + 40027 λ − λ − α τ − λ − λ + 377( λ − α τ + 3200081 20 λ − λ + 6387 λ − λ − α τ + . . . + rightmoving (1 . τ and α by τ and α in the obviousway. The leading term in (1.1) is the usual BTZ result. We note that the factor of λ − W N minimal model coset CFT SU ( N ) k ⊕ SU ( N ) SU ( N ) k +1 (1 . N, k → ∞ , λ ≡ Nk + N fixed (1 . λ ], along with some additional scalarfields that will play no role in the present discussion. Evidence for this proposal, based onsymmetries, RG flows, and the perturbative spectrum, is discussed in [13,14,15,16].To make contact with our black hole result, we should consider the partition functionof this theory with the insertion of a spin-3 chemical potential, Z CF T ( τ, α ; τ , α ) = Tr h e π i ( τ ˆ L + α ˆ W− τ ˆ L− α ˆ W ) i (1 . τ, α → α/τ fixed. This is aversion of the Cardy limit, generalized to include the higher spin chemical potential. Inthis limit, the duality conjecture asserts that (1.1) and (1.4) should agree.While it should be possible to test this prediction for general λ , in this paper we willonly carry out the CFT computation for the special values λ = 0 ,
1. The reason why thesevalues are more tractable is as follows. In general, the symmetry algebra controlling thecoset theory in the ’t Hooft limit is believed to be the infinite dimensional algebra W ∞ [ λ ].This needs to be so in order for the duality conjecture to be true — for instance, the W ∞ [ λ ] algebra is the asymptotic symmetry algebra of hs[ λ ] gravity on AdS [14,28,29,30]— but independent evidence is also available [14,15]. At λ = 0 , λ = 1, after a change of basis, the algebra turns into the linear algebra W PRS ∞ [34].Importantly for us, this algebra can be represented in terms of a collection of free bosons,with the higher spin currents being quadratic in the bosons [35,36,37]. Since the bosonsare free, we can of course compute (1.4) exactly for this theory. If we make the plausibleassumption (justified in more detail in the text) that this free boson theory should sharethe same high temperature partition function as the coset theory at λ = 1, then we arriveat the striking prediction that our black hole result should match a certain free bosonpartition function. Up to the order that we have checked, this turns out to be correct: wefind precise agreement with (1.1) at λ = 1!An analogous story holds at λ = 0, but now in terms of free fermions. At λ = 0 the W ∞ [ λ ] algebra is related to the algebra W ∞ [38] by a constraint that removes the spin-1current. Since the W ∞ algebra can be represented by free fermions [39], we can computeits partition function with the spin-1 constraint imposed. We then find precise agreementwith (1.1) at λ = 0.We view these results as providing strong evidence for the validity of our rules for treat-ing black holes in higher spin gravity, and for applying them to the conjecture of Gaberdieland Gopakumar. Further tests along these lines are clearly possible, most obviously by In general, one might wonder whether such traces are convergent. In the following we willbe considering perturbation theory in α and α , and such issues will not arise. λ , and pushing the comparison to higher (ide-ally all) orders in α . Another useful generalization would be to turn on additional chemicalpotentials. More ambitiously, it seems reasonable to hope that these comparisons will leadto a deeper understanding of how the duality is working at a fundamental level.The remainder of this paper is organized as follows. In section 2, after reviewingthe Chern-Simons formulation of higher spin gravity, we present the rules for constructinghigher spin black hole solutions. These rules are applied to the hs[ λ ] theories in section 3,and the black hole partition function is computed. In section 4 we compute the partitionfunctions for free bosons and fermions, and demonstrate agreement with the bulk result for λ = 0 ,
1. Section 5 contains a discussion of the implications of our results for the AdS/CFTcorrespondence. Appendix A gives the hs[ λ ] structure constants, and in appendix B wedisplay certain holonomy equations in detail.
2. Constructing black holes in higher-spin gravity
We begin with a review of the Chern-Simons formulation of D=2+1 gravity, alongwith the rules developed in [17,18] for constructing black hole solutions. These rules applyto any Chern-Simons formulation of gravity in which the connections take values in a Liealgebra that contains SL(2,R).
It was discovered over two decades ago that Einstein gravity with a negative cosmo-logical constant can be re-written as a SL(2,R) × SL(2,R) Chern-Simons theory [40,41] .With 1-forms ( A, ¯ A ) taking values in the Lie algebra of SL(2,R), the action is S = S CS [ A ] − S CS [ A ] (2 . S CS [ A ] = k π Z Tr (cid:18) A ∧ dA + 23 A ∧ A ∧ A (cid:19) (2 . k is related to the Newton constant G and AdS radius l as k = l G (2 . F = dA + A ∧ A = 0 , F = dA + A ∧ A = 0 (2 . A, A ) to lie in some other Lie algebra besides SL(2,R), whichwe will denote G . Doing so is equivalent to coupling some set of higher spin fields toEinstein gravity, where the rank of G determines the number of higher spin fields. Taking4 = SL(N,R), for example, one has a theory of Einstein gravity coupled to a tower ofsymmetric tensor fields of spins s = 3 , , . . . , N . Taking G to be an algebra of infiniterank introduces an infinite tower of such spins; the details of the theory depend on whichalgebra one chooses. In all of these cases, one recovers Einstein gravity upon restrictionto a SL(2,R) × SL(2,R) subalgebra of
G × G . Note that in general a given G admits manyinequivalent embeddings of SL(2,R); this leads to the appearance of multiple AdS vacuain the theory [18].In this paper, we will take G to be the infinite-dimensional higher spin algebra hs[ λ ],which appears in higher spin contexts old [21,34] and new [13,14,28,29,30]. Our goal willbe to construct a black hole solution in such a theory with higher spin charges turned on.In preparation, we review the spin-3 black hole [17,18] of the G = SL(3,R) theory, whichpossesses only the spin-3 field in addition to the graviton. The spacetime interpretation and asymptotic symmetries of the SL(3,R) × SL(3,R)Chern-Simons theory were treated in detail in [29], following which [17,18] discoveredsmooth, asymptotically AdS , black hole solutions with nonzero spin-3 charge and con-sistent thermodynamics. We refer to those papers for details, here extracting only theessential lessons about how to make a spin-3 black hole.We start from the simple fact that the ordinary BTZ black hole is a solution of theSL(3,R) × SL(3,R) theory. Denoting a SL(2,R) subalgebra of generators as { L ± , L } , theBTZ solution in Chern-Simons language is A = (cid:16) e ρ L − πk L e − ρ L − (cid:17) dx + + L dρA = − (cid:16) e ρ L − − πk L e − ρ L (cid:17) dx − − L dρ (2 . ρ, x ± ) are the spacetime coordinates, with x ± = t ± φ . ( L , L ) are the conservedcharges carried by the black hole, i.e. linear combinations of the mass and angular mo-mentum. To write down the metric g µν and spin-3 field ϕ µνγ , we introduce a generalizedvielbein e and spin connection ω as A = ω + e , A = ω − e (2 . e and ω in a basis of 1-forms dx µ , the spacetime fields are identified as g µν = 12 Tr( e µ e ν ) , ϕ µνγ = 13! Tr( e ( µ e ν e γ ) ) (2 . ϕ µνγ = 0.5he Euclidean BTZ black is obtained by taking dx + = dz and dx − = − dz . Toavoid a conical singularity at the horizon we need to make the identification ( z, z ) ∼ =( z + 2 πτ, z + 2 πτ ), with L = − k πτ , L = − k π ¯ τ (2 . β , and angular velocity of the horizon, Ω, arethen given by τ = iβ + iβ Ω2 π , ¯ τ = − iβ + iβ Ω2 π (2 . ω = 2 π ( τ A + − ¯ τ A − ) (2 . ω is given by its two independent eigenvalues, or equivalently by the values of Tr( ω ) andTr( ω ). For the BTZ solution we computeTr( ω ) = − π , Tr( ω ) = 0 (2 . W , W ). Since black holes represent states ofthermodynamic equilibrium, we of course also need to turn on the corresponding conjugatepotentials ( µ, ¯ µ ). Just as in the BTZ case smoothness at the horizon fixed the relations(2.8) between the spin-2 charges and potentials, here also we expect that smoothnesswill fix a relation between the spin-3 charges and their conjugate potentials. The mainsubtlety, elaborated on in detail in [17,18], is that one cannot impose smoothness by naivelyexamining the local geometry at the horizon, since this local geometry, and even the veryexistence of the horizon, is not SL(3,R) × SL(3,R) gauge invariant.The primary lesson of [17,18] is that one should consider the equations (2.11) tobe the gauge-invariant characterization of a smooth horizon for any solution of theSL(3,R) × SL(3,R) theory. In a theory of flat connections, the holonomy captures thephysics encoded in a given connection, and demanding that the time circle holonomy takeits gauge-invariant value as fixed by the BTZ metric enforces consistency via fixing smooth-ness at the origin of the Euclidean plane. Conveniently, this does not require passage tothe metric-like fields, as in (2.7): given some candidate connection for a spin-3 black hole,we fix the charges by fixing the time circle holonomy.There are two pieces of evidence for the validity of this proposal. As noted above, ina generic gauge the connection for a spin-3 black hole may correspond to a metric with noevent horizon. However, [18] showed that if (and plausibly only if) the holonomy conditions(2.11) are satisfied, somewhere on the gauge orbit of this connection lies a metric with a6ompletely smooth event horizon. This is the sense in which we can meaningfully referto such solutions as black holes.Furthermore, the holonomy prescription guarantees a sensible thermodynamics. Be-fore enforcing these conditions, the solution is a function of the chemical potential µ , thespin-3 charge W , the leftmoving momentum L and the inverse temperature τ , along withall barred partners. We want to think of the black hole as a saddle point contribution toa partition function of the form Z ( τ, α ; ¯ τ , ¯ α ) = Tr[ e π i ( τ L + α W− ¯ τ L− ¯ α W ) ] (2 . L ∼ ∂Z∂τ and
W ∼ ∂Z∂α , Z ( τ, α ; ¯ τ , ¯ α )will exist only if the charge assignments obey the integrability condition ∂ L ∂α = ∂ W ∂τ (2 . α = ¯ τ µ , ¯ α = τ ¯ µ (2 . With this example in mind, we turn to the case of a higher spin theory built upon anarbitrary Lie algebra G which contains a SL(2,R) subalgebra. The prescription for buildingsmooth black holes with higher spin charge is as follows:1. Write down a BTZ solution.2. Compute the BTZ time circle holonomy eigenvalues.3. Write down a flat connection that includes nonzero chemical potentials for some chosenset of higher spin charges.4. Fix the charges in the solution by demanding that the holonomy of the solution aroundthe time circle agrees with that of BTZ.The resulting solution will represent a black hole in the sense described above. Notethat if G is of infinite rank, there will be an infinite number of holonomy constraints.Note also that the these solutions are not in general gauge equivalent to BTZ, since theholonomies around the angular circle will differ. Strictly speaking, this was only shown in the case of static solutions.
7n the above algorithm, step 3 is stated the least explicitly. Fortunately, our spin-3example suggests a straightforward way to find the relevant connections. To explain this,consider the explicit connection used in [17,18] A = (cid:16) e ρ L − πk L e − ρ L − − π k W e − ρ W − (cid:17) dx + + µ (cid:16) e ρ W − π L k W + 4 π L k e − ρ W − + 4 π W k e − ρ L − (cid:17) dx − + L dρ (2 . A . The corresponding metric has no event horizon, butwhen the holonomy conditions are obeyed it is gauge-equivalent to one that does [18].Written in the form (2.15) the solution appears rather complicated, but in fact thestructure is quite simple. First, following [29] we note that we can write A = b − ab + b − db (2 . b = e ρL , and a is obtained from A by setting ρ = dρ = 0. In terms of a , the flatnessequations are simply [ a + , a − ] = 0. To exhibit flatness we need only observe that a − = 2 µ (cid:20) ( a + ) −
13 Tr( a + ) (cid:21) (2 . A + corresponds to choosing the “highest weight gauge”. Namely, if weassume that A + grows as e ρ , then by a gauge transformation it can always be put into theform in (2.15) [29]. Finally, as shown in [17] by a Ward identity analysis, the µe ρ W termin A − gives rise to a chemical potential µ conjugate to spin-3 charge.This discussion suggests a simple way to write down solutions that incorporate chem-ical potentials for higher-spin charges for any G : to turn on potentials µ s for fields of spin s , simply take A + = A BT Z + + (higher spin charges) A − ∼ X s µ s h ( A + ) s − − trace i A ρ = L (2 . G . L is the diagonal element of the SL(2,R) embedding into G used to construct the BTZsolution. As usual, the terms in A − incorporate the sources, and those in A + encodethe charges. Exactly which charges one must turn on in order to have a consistent solu-tion depends on the theory in question, and is determined by solution of the holonomyequations. The generators { W ± , W ± , W } transform in the 5-dimensional representation under theadjoint action of { L ± , L } . Also, as in [17,18] we are here using a representation of the SL(3,R)generators in terms of 3 ×
8s already emphasized, the metric derived from (2.18) may not possess a horizon, butbased on our study of the SL(3,R) theory we expect that there exists another connectionlying on the same gauge orbit that does yield a black hole metric. Finding the explicitgauge transformation will typically be quite involved and G -dependent, but for purposesof interpretation, we merely require its existence.
3. The W ∞ [ λ ] black hole As a last step before writing down the black hole solutions, we review the features ofhs[ λ ] that we will need. In particular, we introduce an associative multiplication known asthe “lone-star product” [33], the antisymmetric part of which yields the hs[ λ ] Lie algebra. λ ] from an associative multiplication The hs[ λ ] Lie algebra is spanned by generators labeled by a spin and a mode index.We use the notation of [14], in which a generator is represented as V sm , s ≥ , | m | < s (3 . V sm , V tn ] = s + t − X u =2 , , ,... g stu ( m, n ; λ ) V s + t − um + n (3 . s = 2 form an SL(2,R) subalgebra, and the remaining generatorstransform simply under the adjoint SL(2,R) action as[ V m , V tn ] = ( m ( t − − n ) V tm + n (3 . λ = 1 /
2, this algebra is isomorphic to hs(1,1), the commutator of which canbe written as the antisymmetric part of the Moyal product. Similarly, the general λ commutation relations (3.2) can be realized as[ V sm , V tn ] = V sm ⋆ V tn − V tn ⋆ V sm (3 . V sm ⋆ V tn ≡ s + t − X u =1 , , ,... g stu ( m, n ; λ ) V s + t − um + n (3 . g stu ( m, n ) = ( − u +1 g tsu ( n, m ) (3 . u drop out of the commutator, leaving (3.2). In the remainder of thepaper we may resort to the shorthandΓ ⋆ Γ ⋆ . . . ⋆ Γ | {z } N ≡ (Γ) N (3 . λ ]-valued element Γ.Formally, V is the identity element. Thus, to extract the trace from a product ofgenerators, one picks out the u = s + t − V sm V tn ) ∝ g sts + t − ( m, n ; λ ) δ st δ m, − n (3 . V sm V s − m ) = 24( λ − g ss s − ( m, − m ; λ ) (3 . V V − ) = − , Tr( V V ) = 2 (3 . λ ] Lie algebra is that when λ = N for integer N ≥ s > N to zero (i.e. factor out the ideal of theLie algebra), and the algebra reduces to SL(N,R). This implies a similar truncation of theboundary symmetry: that is, W ∞ [ N ] = W N upon constraining all fields of spin s > N tovanish. Factoring out the ideal is automatic on the level of the trace:Tr( V sm V s − m ) ∝ s − Y σ =2 ( λ − σ ) (3 . λ = N ; this will be a useful check for us.Another aspect of the lone star product that we wish to highlight is the followingsimple result for products of the highest weight SL(2,R) generator:( V ) s − = V ss − (3 . A − from that of A + . 10n what follows, we work with a flat connection a (and, implicitly, ¯ a ) that has no ρ -dependence nor ρ component, as in (2.17), by writing A = b − ab + b − dbA = b ¯ ab − + bdb − (3 . b = e ρV (3 . b of a generator with mode index m produces a factor e mρ . We now follow the prescription described in section 2.3 for constructing the higherspin black hole. In the hs[ λ ] theory, the BTZ black hole has the connection a + = V + 14 τ V − a − = 0 (3 . V ± are SL(2,R) elements. The BTZ holonomy can beencoded in the infinite set of tracesTr( ω nBT Z ) , n = 2 , , . . . (3 . ω BT Z = 2 πτ (cid:18) V + 14 τ V − (cid:19) (3 . n traces vanish. The lowest even- n traces areTr( ω BT Z ) = − π Tr( ω BT Z ) = 8 π λ − ω BT Z ) = − π λ − λ + 31) (3 . W ∞ [ λ ] black hole Our ansatz for a black hole with spin-3 chemical potential is a + = V − π L k V − − N ( λ ) π W k V − + Ja − = µN ( λ ) (cid:18) a + ⋆ a + − π L k ( λ − (cid:19) (3 . J = J V − + J V − + . . . (3 . N ( λ ) is a normalization factor, N ( λ ) = s λ −
4) (3 . s > Suppressing the dependence on barred quantities, we think of this black hole as asaddle point contribution to the partition function Z ( τ, α ) = Tr h e π i ( τ L + α W ) i (3 . α = τ µ (3 . τ is the modular parameter of the boundary torus, defined via the identification( z, z ) ∼ = ( z + 2 πτ, z + 2 πτ ), with x + = z, x − = − z . This will once again be justified uponsolving the holonomy equations, as the charges will satisfy the integrability condition ∂ L ∂α = ∂ W ∂τ (3 . a + is the asymptotically AdS connection written in the “highest weight gauge” that was used to reveal the asymptotic W ∞ [ λ ] symmetry in [28,29]. The component a − is a traceless source term that deformsthe UV asymptotics: by (3.12), a − = µN ( λ ) V + (subleading) (3 . λ ] black hole has some properties that are quitedifferent from its SL(3,R) counterpart. First, there is an infinite set of holonomy constraintsto solve, corresponding to enforcing smoothness across the horizon of the metric and higherspin fields. Furthermore, solution of these constraints demands that all higher-spin chargesare turned on. This is due to the structure of the W ∞ [ λ ] algebra. For instance, the WW OPE has a term W ( z ) W (0) ∼ . . . + µJ (0) z + . . . (3 . To be clear, there is no pathology for λ ≤
2: we could easily rescale generators to eliminateany troublesome factors of λ − .4. Holonomy For the W ∞ [ λ ] black hole ansatz (3.19), the holonomy matrix is ω = 2 π " τ a + − αN ( λ ) (cid:18) a + ⋆ a + − π L k ( λ − (cid:19) (3 . a + and N ( λ ) as in (3.19) and (3.21), respectively. The holonomy constraints areTr( ω n ) = Tr( ω nBT Z ) , n = 2 , , . . . (3 . α . We assume a perturbative expansionof the form L = L + α L + . . . W = α W + α W + . . .J = α J (2)4 + α J (4)4 + . . .J = α J (3)5 + α J (5)5 + . . . (3 . µ sourcesthe spin-4 current at O ( µ ), the spin-5 current at O ( µ ), and onwards, which this ansatzincorporates. Note the parity under sign flip of α .To exhibit the structure of the holonomy equations, and to set up our perturbativesolution, we write the terms that contribute at lowest perturbative order for each chargethat appears in a given equation, ignoring all of the coefficients and displaying just thefirst four equations: n = 2 : C (2) BT Z = L + α W + α J + . . .n = 3 : C (3) BT Z = α L + W + αJ + α J + α J + . . .n = 4 : C (4) BT Z = L + α WL + J + αJ + α J + α J + α J + . . .n = 5 : C (5) BT Z = α L + WL + αJ L + J + αJ + α J + α J + α J + α J + . . . (3 . C ( n ) BT Z stand for the BTZ holonomies (3.18) that are of course of order α . The “ . . . ”denote terms that contribute at higher perturbative order (for instance, an α L term at n = 2). At each value of n , two more charges enter at ever-higher orders in α . In appendixB we write out the all-order holonomy equations up to n = 4, with spins J and higher setto zero for simplicity.In the case that the gauge algebra is SL(N,R), as obtained by setting λ = N , thesystem of equations terminates at n = N . For the SL(3,R) theory studied in [17,18] thisallowed the holonomy equations to be solved exactly as the solution of a cubic equation.13or general λ we instead must proceed perturbatively. In examining the structure of theequations (3.30) one might be concerned by the fact that at even(odd) orders in α , all even(odd) n equations contribute. This implies that the system is highly overconstrained,infinitely so, in fact; nevertheless it turns out that there is a consistent solution that satisfiesthe integrability condition, at least as far as we have checked.We solve through O ( α ). Combining (3.29) and (3.30), we see that we only need workup to n = 6. The solution is L = − k πτ + 5 k πτ α − k πτ λ − λ − α + 2600 k πτ λ − λ + 377( λ − α − k πτ λ − λ + 6387 λ − λ − α + . . . W = − k πτ α + 200 k πτ λ − λ − α − k πτ λ − λ + 377( λ − α + 32000 k πτ λ − λ + 6387 λ − λ − α + . . .J = 359 τ λ − α − τ λ − λ − α + 28009 τ λ − λ + 3189( λ − α + . . .J = 100 √ τ λ − / α − √ τ λ − λ − / α + . . .J = 1430081 τ λ − α + . . . (3 . L , then that of W , then the subleading term in L , then the leadingterm in J , and so on. From the solutions of L and W we readily confirm that theintegrability equation (3.24) is obeyed, although it has to be said that at our current levelof understanding this appears as a minor miracle. We take this to be powerful evidencethat the holonomy prescription is the correct one for defining higher spin black holes withconsistent thermodynamics.To obtain the partition function we can integrate either one of the equations ∂ ln Z ( τ, α ) ∂τ = 4 π i L , ∂ ln Z ( τ, α ) ∂α = 4 π i W (3 . Z ( τ, α ) = iπk τ " − α τ + 40027 λ − λ − α τ − λ − λ + 377( λ − α τ + 3200081 20 λ − λ + 6387 λ − λ − α τ + . . . (3 . S can be obtained by applying standard thermodynamics: S = ln Z ( τ, α ) − π i ( τ L + α W − τ L − α W ) (3 . α the the entropy is A/ G , where A is the areaof the BTZ horizon, but at higher orders in α no geometric interpretation is evident. Ignoring the factors of λ − J s = τ − s ∞ X n =0 (cid:16) ατ (cid:17) n + s − P ( s ) n ( λ ) , s ≥ . W ≡ J . The degree n polynomials P ( s ) n ( λ ) have zeroes that do notcoincide with those of other values of n or s , and so are unlikely to carry any significance.A first check on our result (3.31) is the λ -independence of the first correction to W and, by integrability, to L . This can be understood as following from the universal leadingcoefficient of the WW OPE, which in our normalization is W ( z ) W (0) ∼ − kπ z + . . . (3 . λ ] trace automatically mods out the effects of higherspin generators upon taking λ = N for integer N ≥
3. So computing the holonomy inhs[ λ ] and then evaluating at λ = N is identical to computing the holonomy in the SL(N,R)theory from the start.To verify this, we have embedded the black hole with spin-3 chemical potential in thetheories with Lie algebras G =SL(3,R), SL(4,R), SL(5,R). The results of these investiga-tions match (3.31) exactly (modulo the non-existence of some of the J charges in thesecases). Furthermore, we have confirmed that the holonomy equations themselves reduceto those of SL(N,R) non-perturbatively (see appendix B for an example).Another useful check is to consider hs[ ]. In this case, the gauge algebra can berepresented in terms of a Moyal product, with generators being even degree polynomialsin two spinor variables [21]. All computations can then be carried out in this framework,and the results precisely agree with (3.31) at λ = 1 / The sign convention adopted here differs from that in most CFT references, but turns out tobe more convenient. . λ = 0 , : comparison to CFT Recall that W ∞ [ λ ] is the asymptotic symmetry of the AdS vacuum of the hs[ λ ] theory,and possesses hs[ λ ] as its wedge subalgebra in the c → ∞ limit, as discussed in [14]. Inanticipation of an application of our results to the holographic realm, we switch gearsand study two CFT realizations of W ∞ [ λ ] symmetry. One is a theory of free bosons at λ = 1, and the other, a theory of free fermions at λ = 0. We compute their exact partitionfunctions in the presence of a spin-3 chemical potential; the perturbative expansions match(3.33). We defer a discussion of why this should be, and its intriguing implications, to thenext section.For easy reference, the gravity results for the leftmoving part of the partition functionare λ = 1 : ln Z ( τ, α ) = iπk τ − iπk α τ + 400 iπk α τ − iπk α τ + 10400000 iπk α τ + · · · λ = 0 : ln Z ( τ, α ) = iπk τ − iπk α τ + 350 iπk α τ − iπk α τ + 5839250 iπk α τ + · · · (4 . λ = 1 : Free bosons A theory of D free complex bosons has global W ∞ [1] symmetry with central charge c = 2 D [35,36,37]. The symmetry algebra is also known [33] as W PRS ∞ , and can be writtenin a linear basis, unlike the W ∞ [ λ ] algebra for other values of λ .The complex bosons have OPE ∂φ i ( z ) ∂φ j ( z ) ∼ − δ ij ( z − z ) , i, j = 1 , , . . . D (4 . T = − ∂φ i ∂φ i W = ia (cid:16) ∂ φ i ∂φ i − ∂φ i ∂ φ i (cid:17) (4 . a is some normalization constant. W is Virasoro primary, as it should be to matchwith the spin-3 current of our bulk theory. The other higher spin currents are also quadraticin the scalars but include more derivatives; the linearity of the symmetry algebra followsfrom the quadratic nature of these currents.To fix a , we compute the leading part of the WW OPE: W ( z ) W (0) ∼ − a Dz + · · · (4 . W ( z ) W (0) ∼ − kπ z (4 . a = r k π D (4 . D for k according to c = 6 k = 2 D , we have a = r π (4 . w = σ + iσ , related to the plane coordinate z as z = e − iw . We suppress the i indices forthe time being. We write ∂φ ( w ) = − X m β m e imw , ∂φ ( w ) = − X m β m e imw (4 . β m , β n ] = mδ m, − n = [ β m , β n ] (4 . T = − ∞ X m =1 (cid:16) β − m β m + β − m β m (cid:17) + k . k/ L is related to the constant part of the stress tensor as L = − π T (4 . L = 12 π ∞ X m =1 (cid:16) β − m β m + β − m β m (cid:17) (4 . W . We get, after normal ordering, W = 2 a ∞ X m =1 m (cid:16) β − m β m − β − m β m (cid:17) (4 . | n m , n m i = ( β − m ) n m ( β − m ) n m | i (4 . L| n m , n m i = 12 π m ( n m + n m ) | n m , n m iW| n m , n m i = 2 am ( n m − n m ) | n m , n m i (4 . Z ( τ, α ) = Tr h e π i ( τ L + α W ) i (4 . Z ( τ, α ) = − D ∞ X m =1 h ln (cid:16) − e πiτm − π iaαm (cid:17) + ln (cid:16) − e πiτm +8 π iaαm (cid:17)i (4 . τ →
0, we can convert the sum to an integral. Thisgives ln Z ( τ, α ) = − ik πτ Z ∞ dx h ln (cid:16) − e − x + iaατ x (cid:17) + ln (cid:16) − e − x − iaατ x (cid:17)i (4 . D = 3 k .It is straightforward to expand in powers of α and do the integrals:ln Z ( τ, α ) = iπk τ − iπk α τ + 400 iπk α τ − iπk α τ + 10400000 iπk α τ + · · · (4 . λ = 0 : Free fermions A theory of D free complex fermions furnishes W ∞ symmetry with central charge c = D [39]. The W ∞ algebra has spins s = 1 , , , . . . , and is related to W ∞ [0] by aconstraint that eliminates the spin-1 current. In the following we will proceed by using achemical potential to demand that the spin-1 charge is set to zero in the partition function.The fermions have OPE ψ i ( z ) ψ j ( z ) ∼ δ ij z − z (4 . T = − ψ i ∂ψ i − ψ i ∂ψ i (4 . b thatwe will fix in a moment, W = ib (cid:0) ∂ ψ i ψ i − ∂ψ i ∂ψ i + ψ i ∂ ψ i (cid:1) (4 . T W OPE contains an extra term ofthe form
J/z where J is the spin-1 current. From this point of view it is clear that tocompare with the bulk we need to set J = 0 so that W will appear as effectively primary.Now we consider the leading part of the WW OPE, which is W ( z ) W (0) ∼ − Db z + · · · (4 . D complex fermions. To match to our usual normalization weshould take b = r π (4 . D = c = 6 k .The mode expansion on the cylinder is ψ ( w ) = e − iπ X m b m e imw , ψ ( w ) = e − iπ X m b m e imw (4 . { b m , b n } = δ m, − n (4 . T ( w ) = − ∞ X m =1 m (cid:0) b − m b m + b − m b m (cid:1) (4 . L = 12 π ∞ X m =1 m (cid:0) b − m b m + b − m b m (cid:1) (4 . W = − b ∞ X m =1 m (cid:0) b − m b m − b − m b m (cid:1) (4 . | n m , n m i = ( b − m ) n m ( b − m ) n m | i (4 . We suppress the i indices. L| n m , n m i = 12 π m ( n m + n m ) | n m , n m iW| n m , n m i = − bm ( n m − n m ) | n m , n m i (4 . Q ∼ R ψψ . Ourprecise definition of the charge operator is Q | n m , n m i = ( n m − n m ) | n m , n m i (4 . Q = 0 as an exact condition on states or as an expectation value; the latter is moreconvenient since it can be imposed by including a chemical potential for Q and tuning itappropriately. The partition function including a chemical potential for Q is Z ( τ, α, γ ) = Tr h e π i [ τ L + α W ]+ iγQ i (4 . Z ( τ, α, γ ) = D ∞ X m =1 h ln (cid:16) e (2 πiτm − π ibαm + iγ ) (cid:17) + ln (cid:16) e (2 πiτm +24 π ibαm − iγ ) (cid:17)i (4 . Z ( τ, α, γ ) = 3 ikπτ Z ∞ dx h ln (cid:16) e − x + ibατ x + iγ (cid:17) + ln (cid:16) e − x − ibατ x − iγ (cid:17)i (4 . γ by demanding charge neutrality. The charge is obtained by differentiatingwith respect to γ and so we need0 = Z ∞ dx (cid:20) e − x − iǫx − iγ + 1 − e − x + iǫx + iγ + 1 (cid:21) (4 . ǫ = 6 bατ (4 . γ = − π ǫ + 16 π ǫ − π ǫ + 1254656 π ǫ + · · · (4 . ǫ , and compute the integrals. After insertingthe value of b given above, we findln Z ( τ, α ) = iπk τ − iπk α τ + 350 iπk α τ − iπk α τ + 5839250 iπk α τ + · · · (4 . . Implications for higher spin AdS /CFT duality We now consider what lessons can be drawn from the agreement between our bulkgravity computations and those for free bosons and fermions. For this discussion, let usmake the assumption that the agreement will persist to all order in α .Symmetry obviously plays a powerful role in determining these partition functions.The most likely explanation for why we see agreement is that the answer is fixed bysymmetry. On the bulk side, our black hole solutions just involve the non-propagatingbulk fields described by the Chern-Simons action, and not the additional scalar fields thatarise in the context of the conjecture [13]. Since the topological sector is what gives riseto the asymptotic symmetry algebra, it seems plausible that the physical properties ofsolutions that lie in this sector are fixed by symmetry.On the CFT side, a symmetry argument would probably proceed along the followinglines. The partition functions we compute are determined, at order α n , by the n-pointcorrelation functions of spin-3 currents on the torus. We are interested in the high temper-ature behavior of these correlation functions. Performing modular transformations termby term, the leading high temperature behavior will be related to correlation functions atlow temperature, which are evaluated on the infinite cylinder, or equivalently the plane.Finally, the correlation functions of spin-3 currents on the plane can be computed from theOPEs. Thus, given the OPEs, we expect that we should be able to compute the partitionfunction in the high temperature limit, and it should agree with our gravity result.Nothing in this argument depends on considering the special cases λ = 0 , λ we expectagreement between our black hole partition function and that for the coset CFT (1.2) inthe high temperature regime. From this point of view, it would be especially interestingto try to understand and match the subleading asymptotics.As was already mentioned in the introduction, our final result should be thought ofas a Cardy formula for CFTs with W ∞ [ λ ] symmetry and with large central charge.Although we have argued that our successful matching of partition functions in termsof free fermions and bosons is a consequence of symmetry, it is interesting to note that for λ = 0 it is believed that the theory (1.2) is in fact fully equivalent to free fermions with asinglet constraint. It would therefore be very interesting to carry out further tests of theAdS/CFT duality [13] at λ = 0.We conclude with one final simple observation. Both for λ = 0 and λ = 1, in the freefermion/boson theories, there are natural candidates for operators dual to the scalar fieldsthat appear on the bulk side of the duality [13]. These bulk scalars are dual to spinlessoperators in the CFT of dimension ∆ = 1 ± λ . At λ = 0 we have the free fermion operator ψ ( z ) ˜ ψ ( z ), and at λ = 1 we have the free boson operator ∂φ ( z ) ∂φ ( z ), both of which havethe appropriate dimension. We thank Matthias Gaberdiel for discussions of these matters. cknowledgments This work was supported in part by NSF grant PHY-07-57702. We thank MartinAmmon, Alejandra Castro, Matthias Gaberdiel, Michael Gutperle, Finn Larsen, and AlexMaloney for discussions. P.K. thanks the Aspen Center for Physics for hospitality duringthe completion of this work.
Appendix A. The hs[ λ ] Lie algebra The hs[ λ ] structure constants are given as g stu ( m, n ; λ ) = q u − u − φ stu ( λ ) N stu ( m, n ) (A.1)where N stu ( m, n ) = u − X k =0 ( − k (cid:18) u − k (cid:19) [ s − m ] u − − k [ s − − m ] k [ t − n ] k [ t − − n ] u − − k φ stu ( λ ) = F " + λ , − λ , − u , − u − s , − t , + s + t − u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (A.2)We make use of the descending Pochhammer symbol,[ a ] n = a ( a − ... ( a − n + 1) (A.3) q is a normalization constant that can be scaled away by taking V sm → q s − V sm . As inmuch of the existing literature, we choose to set q = 1 / φ stu (cid:18) (cid:19) = φ st ( λ ) = 1 N stu ( m, n ) = ( − u +1 N tsu ( n, m ) N stu (0 ,
0) = 0 N stu ( n, − n ) = N tsu ( n, − n ) (A.4)The first three of these imply, among other things, the isomorphism hs[ ] ∼ = hs(1,1); thatthe lone star product can be used to define the hs[ λ ] Lie algebra; and that all zero modescommute. 22 ppendix B. Holonomy equations with J = 0Here we present the holonomy equationsTr( ω n ) = Tr( ω nBT Z ) (B.1)for the black hole connection (3.19), up to n = 4. In the interest of clarity and space, weonly include charges up to J . We find n = 2 : 0 = α J (cid:0) k ( λ − (cid:1) − π α L − πατ k W− πτ k L − k n = 3 : 0 = αJ (cid:16) k ( λ − λ − (cid:2) πα L ( λ −
16) + 9 kτ ( λ − (cid:3)(cid:17) − α π h W k (5 λ − λ + 264) + 256 L π ( λ − λ − i − α WL π τ k ( λ − λ − − α L π τ k ( λ − − W πτ k ( λ − n = 4 : 0 = α J (cid:16) k ( λ − λ − (cid:2) λ − λ + 21707 λ − (cid:3)(cid:17) − J (cid:16) k ( λ − λ − h α L π (7 λ − λ + 1788)+ 8400 α W πτ k (5 λ − λ + 636)+ 23760 α L πτ k ( λ − λ −
11) + 99 τ k ( λ − i(cid:17) + 665600 α L π h W k ( λ − λ − λ + 636)+ 352 L π ( λ − λ − λ + 100) i + 131788800 α WL π τ k ( λ − h λ − λ + 244 i + 137280 α π τ k ( λ − h L π ( λ − λ − W k (5 λ − λ + 264) i + 823680 α WL π τ k ( λ − h λ − i − k (3 λ − λ − ( k + 8 π L τ )( k − π L τ ) (B.2)We have organized these equations so as to reveal the J dependence. Comparison to(3.30) reveals the underlying structure discussed in the main text.It is instructive to ask what happens when we take λ = 3. Since this reduces to theSL(3,R) case, in which there are only two independent holonomy equations, one requires23hat the n = 4 equation should vanish on account of the other two, and that any J dependence should drop out of these equations.The latter is evident upon inspection. And indeed, taking λ = 3 reduces the mess of n = 4 to be proportional to the n = 2 equation by a finite factor. Both the n = 2 and n = 3 equations reduce to those in [17] (see e.g. equation (5.14) there).24 eferences [1] M. A. 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