Scalar fields and three-point functions in D=3 higher spin gravity
aa r X i v : . [ h e p - t h ] N ov Scalar fields and three-point functionsin D = 3 higher spin gravity Martin Ammon, Per Kraus and Eric Perlmutter Department of Physics and AstronomyUniversity of California, Los Angeles, CA 90095, USA
Abstract
We compute boundary three-point functions involving two scalars and a gauge fieldof arbitrary spin in the AdS vacuum of Vasiliev’s higher spin gravity, for any deformationparameter λ . In the process, we develop tools for extracting scalar field equations inarbitrary higher spin backgrounds. We work in the context of hs[ λ ] ⊕ hs[ λ ] Chern-Simonstheory coupled to scalar fields, and make efficient use of the associative lone-star productunderlying the hs[ λ ] algebra. Our results for the correlators precisely match expectationsfrom CFT; in particular they match those of any CFT with W ∞ [ λ ] symmetry at largecentral charge, and with primary operators dual to the scalar fields. As this is expectedto include the ‘t Hooft limit of the W N minimal model CFT, our results serve as furtherevidence of the conjectured AdS/CFT duality between these two theories.November 2011 [email protected], [email protected], [email protected] ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Matter fields in Vasiliev gravity . . . . . . . . . . . . . . . . . . . . . . . . . 43. Generalized Klein-Gordon equations in higher spin backgrounds . . . . . . . . . . . 83.1. AdS: Recovering the Klein-Gordon equation . . . . . . . . . . . . . . . . . . 103.2. Chiral higher spin deformations of AdS . . . . . . . . . . . . . . . . . . . . 123.3. Higher spin currents in AdS . . . . . . . . . . . . . . . . . . . . . . . . . 134. Three-point correlators from the bulk . . . . . . . . . . . . . . . . . . . . . . 154.1. Spin-1 example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2. General spin correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.3. The AdS master field C . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.4. Final result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235. Three-point correlators from CFT . . . . . . . . . . . . . . . . . . . . . . . . 246. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.1. Comparison with previous results . . . . . . . . . . . . . . . . . . . . . . 286.2. Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Appendix A. Lightning review of 3D higher spin gravity coupled to scalars . . . . . . . 30A.1. Vacuum solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31A.2. Matter equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Appendix B. hs[ λ ], and Moyal vs. lone-star products . . . . . . . . . . . . . . . . . 34Appendix C. Derivation of (5.14) . . . . . . . . . . . . . . . . . . . . . . . . . . 38Appendix D. Low spin results: C s in AdS . . . . . . . . . . . . . . . . . . . . . . 38
1. Introduction
Higher spin gravity in AdS has been receiving increased attention in recent years forits promise in shedding light on certain big questions in string theory. For one, stringspropagating in an AdS spacetime of string scale curvature are expected to be described bysome theory of interacting higher spins [1,2,3]; while the precise theory remains unknown,the theories developed by Vasiliev and collaborators [4,5,6,7,8,9] appear to be, at the veryleast, sophisticated toy models of such a theory, with a large gauge symmetry and nonlocaldynamics. For another, the higher spin AdS/CFT dualities known thus far involve vector-like boundary theories, that is, CFTs whose central charges scale as N rather than N .The tractability of these vector models encourages the optimistic view that such dualityconjectures may be derivable, allowing us to peer inside the black box of AdS/CFT.The prototype duality proposal is that of Klebanov and Polyakov [10], who conjectureda duality between (a subsector of) Vasiliev’s theory of higher spins on AdS and the large N limit of the 2+1-dimensional O(N) vector model. This proposal was studied early onin [11,12,13], as well as in the recent papers [14,15,16,17,18]. Through a combination ofdirect computation of bulk three-point functions, generalization to n -point functions, andarguments regarding higher spin symmetry breaking and AdS boundary conditions, these1orks provide a powerful body of evidence that the duality is correct, at least as far as thebulk theory is understood.An AdS analog of the Klebanov-Polyakov conjecture is the proposed duality [19]between Vasiliev gravity in D=3 [20] coupled to a pair of complex scalar fields and a ’tHooft limit of the W N minimal model CFT, which has a coset representation SU ( N ) k ⊕ SU ( N ) SU ( N ) k +1 (1 . N, k → ∞ , λ ≡ Nk + N fixed (1 . N CFT partition function [22]; a higher spin black hole partition function at hightemperature and the same quantity in the CFT [23]; and scalar-scalar-higher spin three-point functions at λ = 1 / s = 2 , ,
4, in the ’t Hooft limit [24,25].It is this last piece of evidence upon which we build in the present work, with the aim ofdrawing closer to a perturbative proof of the conjecture a la [15]. There are two immediateways we would like to generalize previous calculations of three-point correlators. The bulkcomputation of [24] was in the “undeformed” Vasiliev theory, that is, at λ = 1 / ν = 0in the language of [20]), and we want to extend this to arbitrary λ . The minimal modelcomputations of [24,25] only treat the low spins, and we want to compute for arbitraryspin.In this paper, we succeed in computing the bulk correlators for arbitrary spin andarbitrary λ , and show that they match precisely with CFT expectations. On the CFTside we proceed under the assumption that the theory has W ∞ [ λ ] symmetry. A separatequestion, not addressed here, is whether this is indeed true of the minimal model CFTsin the ’t Hooft limit; evidence in favor appears in [21,22,25]. Along the way, we presenttechniques for writing generalized bulk scalar wave equations in arbitrary on-shell higherspin backgrounds, which have interesting applications beyond the present context.We now state our result for the three-point correlators. First, in the deformed bulktheory, we calculate the three-point function of two scalar fields and a higher spin field ofarbitrary spin s . Recall that there are two complex scalar fields in the bulk, each with m = λ −
1. Taking one of them to be dual to an operator O and its complex conjugate O , our result for the three-point function in terms of the scalar-scalar two-point functionis hO ± ( z ) O ± ( z ) J ( s ) ( z ) i = ( − s − π Γ( s ) Γ(2 s −
1) Γ( s ± λ )Γ(1 ± λ ) (cid:18) z z z (cid:19) s hO ± ( z ) O ± ( z ) i (1 . − ) quantization of the scalar. Thesame correlator for the other scalar field, dual to an operator e O and its complex conjugate e O , is identical to (1.3) but absent the ( − s prefactor. In the context of the duality [19],these operators should be assigned to take opposite quantization to one another.Let us now mention a few of the main insights that allowed us to compute thesecorrelators relatively easily. First, it is well known that if the scalar fields are set to zerothe Vasiliev theory is equivalent to a Chern-Simons theory with gauge algebra hs[ λ ] ⊕ hs[ λ ].To compute correlators of the type (1.3) we need to couple a free scalar field to this theory.The general rules for incorporating scalar fields into the higher spin theory are complicatedbut known (see [20] and appendix A). However, free scalar field equations can be derivedfrom the elegant equation dC + A ⋆ C − C ⋆ A = 0 (1 . A, A ) denote the hs[ λ ] ⊕ hs[ λ ] gauge fields. As we will explain, C is a “master field”that takes values in the Lie algebra hs[ λ ] supplemented with an identity element, and thescalar is the part of C proportional to the identity. This equation reduces to the Klein-Gordon equation for a scalar of mass m = λ − A and A representing higher spin gauge fields with prescribed asymptotics.Such flat connections can be generated by gauge transformations. Therefore, if we startfrom a solution for the free scalar in AdS and then act with the gauge transformation,we generate a new scalar solution in the presence of the higher spin gauge fields. In thisway, rather than having to first work out the perturbed scalar equation and then solve it,we can generate the solution in one step, which is a huge simplification.On the CFT side, our starting point is the assumption that in the ’t Hooft limit the W N coset CFT has W ∞ [ λ ] global symmetry. While this is unproven, it is a prerequisitefor the duality to hold in the pure gauge sector. Previous calculations [24,25] took thetack of computing the s = 2 , , N in the CFT, and takingthe ’t Hooft limit afterwards; this serves as good evidence that W ∞ [ λ ] really does emergein the ’t Hooft limit, but the complications of finite N are not required if one wants to A second type of scalar, dual to the e O operators noted above, obeys the same equation butwith A and A interchanged. W ∞ [ λ ] — in particular,of its large k wedge subalgebra hs[ λ ] — fixes the correlator. Generalizing results in [21],we show that, in fact, it is enough to reproduce the result (1.3) and the accompanyingresult for the second scalar, providing perfect agreement with the bulk.Our results for bulk and boundary correlators reduce to previous computations [24,25]in the appropriate limits. In all, we find this to be a significant step toward verifying theduality proposal [19].Our techniques will also have application to computing other correlation functions inthese theories. For instance, there is by now a good understanding of higher spin blackholes in D=3 [27,28,23,29,30]; in particular their entropy is known to match that of thedual CFT in the high temperature limit [23]. Using the results of this paper, it is nowquite feasible to compute scalar correlators in the background of a higher spin black holeand compare with CFT.The remainder of this paper is organized as follows. In section 2 we go through themain steps involved in deriving the equation (1.4) from the general formulation of theVasiliev theory. Further details are provided in appendix A. In section 3 we show how towork out the explicit form of the scalar wave equation in the presence of higher spin gaugefields. In section 4, which is the core of the paper, we show how to use gauge invariance togenerate solutions of the scalar wave equations, and then read off the desired correlationfunctions. In section 5 we compute these correlators on the CFT side under the assumptionof W ∞ [ λ ] symmetry, and demonstrate perfect agreement with the bulk. We conclude withsome comments in section 6. Appendix B presents some evidence for the isomorphismbetween the lone-star product and the Moyal product acting on deformed oscillators, inAppendix C we derive a result needed in the text, and Appendix D provides some usefulexplicit expressions for comparison with a formula derived in section 4.
2. Matter fields in Vasiliev gravity
We begin with a review of the formulation of D = 3 higher spin gravity due to Vasilievand collaborators, as presented in [20]. We first recall how to write the gauge sector of thistheory as a hs[ λ ] ⊕ hs[ λ ] Chern-Simons theory, and then show how to introduce linearizedscalar fields in the Chern-Simons language. Seeing as we will not need all of the details ofthe theory’s construction, we present an abridged discussion; the reader who would prefernot to take anything on faith is referred to appendix A.Vasiliev gravity contains one higher spin gauge field for each integer spin s ≥
2, coupledto some number of matter multiplets. There are various ingredients, foremost among thema set of “master fields:” a spacetime 1-form W = W ν dx ν as well as spacetime 0-forms B and S α . Besides the spacetime coordinates x, the generating functions W, B and S α also4epend on auxiliary bosonic twistor variables z α , y α where α = 1 , , as well as on twopairs of Clifford elements: ψ , , and k, ρ . That is, { ψ i , ψ j } = 2 δ ij , kρ = − ρk , k = ρ = 1 (2 . ψ , commute with all other auxiliary variables, k and ρ have the properties ky α = − y α k, kz α = − z α kρy α = y α ρ, ρz α = z α ρ (2 . W encodes the gauge sector, B parameterizes the AdS vacua ofthe theory and is used to introduce propagating matter fields, and S α ensures that thetheory has the correct internal symmetries. The elements { z α , y α , k, ρ } are ingredientsin the realization of these symmetries, and the ψ i are required only when writing downsolutions.Twistor indices are raised and lowered by the rank two antisymmetric tensor ǫ αβ : z α = ǫ αβ z β , z α = z β ǫ βα (2 . ǫ = ǫ = 1 . Functions of the twistors z α , y α are multiplied by the Moyal product: f ( z, y ) ⋆ g ( z, y ) = 1(2 π ) Z d u Z d v e iu α v α f ( z + u, y + u ) g ( z − v, y + v ) (2 . z α , ˜ y α . The ˜ y α , which will be more importantfor our work here, obey deformed oscillator star-commutation relations[˜ y α , ˜ y β ] ⋆ = 2 iǫ αβ (1 + νk ) (2 . λ ] as follows. Defineelements of the algebra to consist of symmetrized, positive even-degree polynomials in ˜ y α .Multiplying these elements using (2.5), and projecting onto k = ±
1, the commutationrelations are those of hs[ λ ], with λ = (1 ∓ ν ). This deformed oscillator algebra plays a central role in the study of AdS vacua inVasiliev theory. It emerges dynamically from the field equations of the full nonlinearsystem of higher spins. The parameter ν encodes the deformation, and in the event that The z α , y α are sometimes referred to as ”oscillators.” More precisely, we consider polynomials of degree two and higher, since the constant termstar-commutes with everything. We also note that one can enlarge hs[ λ ] by including odd powersof the ˜ y α in the polynomials, though our interests here are in purely bosonic Vasiliev theory. = 0, the theory is said to be undeformed. ν also plays two other important roles: itparameterizes a family of inequivalent AdS vacua, and sets the mass of any scalar fieldsthat we introduce to the theory. These connections stem from the structure of the fieldequations, themselves tightly constrained by higher spin gauge invariance.To see how ν appears in connection with the AdS vacua, we examine the field equa-tions. There are five equations in terms of the master fields W, B and S α , and we presenttwo of them here (the others are written in the appendix): dW = W ∧ ⋆WdB = W ⋆ B − B ⋆ W (2 . S α . At the order to which we will beworking in this paper — namely, linear in the scalar fields — the entire effect of S α is thatit forces W and B to be independent of k and ˜ z α . With this in hand, we can proceed byfocussing on (2.6).We first consider solutions with vanishing scalar field. This corresponds to taking aconstant background value for B , B = ν (2 . ν is fixed by the omitted S α equations to be the same parameter asappears in the deformed oscillator expressions.Now, the first equation in (2.6) can be written as a flatness condition for two Chern-Simons gauge fields, each taking values in the Lie algebra hs[ λ ]. In order to see this weintroduce gauge fields A and A by W = −P + A − P − A (2 . A and A are functions of ˜ y α and the spacetime coordinates x µ . Here we haveintroduced the projection operators P ± = 1 ± ψ . P ± ψ = ψ P ± = ±P ± , P ± ψ = ψ P ∓ (2 . dA + A ∧ ⋆A = 0 dA + A ∧ ⋆A = 0 (2 . A and A are taken to be polynomials of positiveeven degree in symmetrized products of ˜ y α , then (2.11) are equivalent to the field equationsof hs[ λ ] ⊕ hs[ λ ] Chern-Simons theory. Since SL(2) is a subalgebra of hs[ λ ], this theory6ncludes ordinary Einstein gravity with a negative comological constant as a consistenttruncation.Before introducing the scalar fields, let us say a bit more about hs[ λ ]. The hs[ λ ] Liealgebra is spanned by generators labeled by a spin index s and a mode index m . We usethe notation of [21], in which a generator is represented as V sm , s ≥ , | m | < s (2 . V sm , V tn ] = s + t −| s − t |− X u =2 , , ,... g stu ( m, n ; λ ) V s + t − um + n (2 . s = 2 form an SL(2)subalgebra, and the remaining generators transform simply under the adjoint SL(2) actionas [ V m , V tn ] = ( m ( t − − n ) V tm + n (2 . λ can be mapped to the parameters of the oscillator formulation as λ = 1 − νk . k = ±
1. When λ = 1 /
2, the theory is undeformed and this algebra is isomorphic tohs(1,1) [31].To summarize what we have found so far, the gauge sector of the Vasiliev theory boilsdown to hs[ λ ] ⊕ hs[ λ ] Chern-Simons theory. This sector of the theory has no propagatingdegrees of freedom.To introduce propagating scalar fields we study a linearized fluctuation of B aroundits vacuum value (2.7), B = ν + C ( x, ψ i , ˜ y α ) (2 . C is taken to have an expansion in even-degree symmetrized products ofthe deformed oscillators ˜ y α . The lowest term in the expansion, with no deformed oscillators,will be identified with the physical scalar field. C obeys d C −
W ⋆ C + C ⋆ W = 0 (2 . C as C = P + ψ C ( x, ˜ y α ) + P − ψ ˜ C ( x, ˜ y α ) (2 . dC + A ⋆ C − C ⋆ A = 0 d e C + A ⋆ e C − e C ⋆ A = 0 (2 . m = λ −
1. More generally, theseequations capture the interaction of the linearized scalars with an arbitrary higher spinbackground. For example, they can be used to study the propagation of a scalar field inthe higher spin black hole of [23].We note that the two equations (2.19) are related by A ↔ A , C ↔ e C . This isinterpreted as a “charge conjugation” operation that flips the sign of all odd spin tensorgauge fields. Another notable feature is that the equations (2.19) are only sensible for A and A on-shell, i.e. satisfying equation (2.11). This can be seen by taking d of theseequations; if the connections are not flat this leads to extra constraints on C and e C withno interpretation in terms of propagating scalar fields.Recapping, we have now reduced the system of equations down to (2.11) and (2.19).These equations describe the propagation of linearized scalar fields in an arbitrary on-shellhigher spin background. They do not capture backreaction of the scalar on the higherspin fields, or self-interactions among the scalars. Neither of these effects is needed for thecomputation of the three-point correlators herein.We now turn to solving these equations, and introducing efficient tools for this purpose.We focus on the C equation; results for e C then follow by charge conjugation.
3. Generalized Klein-Gordon equations in higher spin backgrounds
Starting from dC + A ⋆ C − C ⋆ A = 0 (3 . λ ]-valued connections ( A, A ), we wantto extract the generalized Klein-Gordon equation hiding within.As we said, in the traditional formalism of bosonic Vasiliev theory, the master field C is expanded in deformed oscillators ˜ y α as C = C + C αβ ˜ y α ˜ y β + C αβσλ ˜ y α ˜ y β ˜ y σ ˜ y λ + . . . (3 . C are symmetric in twistor indices.This separates the components of the master field into the physical scalar field, which is thelowest component C , and the remaining components related on-shell to C by derivatives.Plugging (3.2) into (3.1) leads, after much work, to the scalar equations.The most tedious part of this computation is multiplying the deformed oscillators. Weneed to take a pair of symmetrized combinations of oscillators, multiply them, and thenresymmetrize using (2.5). Rather than carrying out this procedure each time, it wouldbe much more convenient if we had a closed-form expression for the multiplication rules.Recall that the Lie algebra obtained via star-commutation of these elements is hs[ λ ]. Now,underlying hs[ λ ] is an associative product, under which the hs[ λ ] Lie bracket becomes thecommutator (2.13). This “lone-star product” is defined as8 sm ⋆ V tn ≡ s + t −| s − t |− X u =1 , , ,... g stu ( m, n ; λ ) V s + t − um + n (3 . g stu ( m, n ; λ ) = ( − u +1 g tsu ( n, m ; λ ) (3 . W ∞ algebras [26], and used morerecently by two of the present authors [23] to compute black hole partition functions inhs[ λ ] gravity.It is then very natural to suspect an isomorphism between the product rules forsymmetrized oscillator combinations and those of the lone-star product. In appendix Bwe present strong evidence at low spins that, indeed, the lone-star product acting on hs[ λ ]generators is isomorphic to the product involving the deformed oscillators, with a specificidentification between generators and oscillator polynomials.Proceeding under this assumption, which will be well justified by the consistency of allresults, provides a major technical simplification. One trades the tedious symmetrizationprocedure for the known and easily manipulated hs[ λ ] structure constants.In this language, we expand the master field C as follows: C = ∞ X s =1 X | m |
1. Anattractive feature of the theory is that the value of the mass is fixed by the gauge algebra.9 .1. AdS: Recovering the Klein-Gordon equation
The AdS connection is constructed out of the spin-2 generators alone, namely thoseforming an SL(2) subalgebra of hs[ λ ]. We work in Euclidean signature in Fefferman-Graham gauge, with radial coordinate ρ and boundary coordinates ( z, ¯ z ). The connectionis A = e ρ V dz + V dρA = e ρ V − d ¯ z − V dρ (3 . ds = dρ + e ρ dzd ¯ z (3 . A and A — the C equations (3.1) are simple towrite. Decomposing along both spacetime and internal hs[ λ ] space, and using the lone-starproduct, one finds V sm,ρ : ∂ ρ C sm + 2 C s − m + C s +1 m g ( s +1)23 ( m,
0) = 0 V sm,z : ∂C sm + e ρ (cid:18) C s − m − + 12 g s (1 , m − C sm − + 12 g s +1)3 (1 , m − C s +1 m − (cid:19) = 0 V sm, ¯ z : ∂C sm − e ρ (cid:18) C s − m +1 − g s ( − , m + 1) C sm +1 + 12 g s +1)3 ( − , m + 1) C s +1 m +1 (cid:19) = 0(3 . | m | < s and ∂ = ∂ z , ∂ = ∂ ¯ z . (Here and henceforth, we suppress the λ -dependenceof the structure constants g stu ( m, n ).)It is now easy to obtain the Klein-Gordon equation. Writing out a handful of equationsat s = 1 ,
2, one finds four that form a closed set for components { C , C , C , C } : V ,ρ : ∂ ρ C + C · λ −
16 = 0 V , ¯ z : ∂C + e ρ C · λ −
16 = 0 V ,z : ∂C + e ρ C + e ρ C − e ρ C · λ −
430 = 0 V ,ρ : ∂ ρ C + 2 C + C · λ − . In this work, we will not need the prescription to pass from Chern-Simons to metric language;suffice it to say that in writing this metric, we have chosen a particular normalization of the hs[ λ ]trace. See [23] for conventions used here. V sm,x µ is a short-hand notation for the component along V sm dx µ . (cid:2) ∂ ρ + 2 ∂ ρ + 4 e − ρ ∂∂ − ( λ − (cid:3) C = 0 (3 . C must have a smooth solution in terms of C . We delay presentation of the solutionfor the full master field C in AdS until section 4, where we will need it to compute thethree-point functions. There, we will also show that for any connection related to AdSby a non-singular gauge transformation, the linearized matter equations (3.1) admit aconsistent solution for the full master field C .Before moving on to higher spin deformations of AdS, let us elucidate the structure of(3.10) and present a systematic strategy for isolating the minimal set of equations neededto solve for C . We wish to highlight a special type of component of C , namely thosewhich are of the form C m +1 ± m and hence have the smallest possible spin for fixed mode m : C , C ± , etc. We call these “minimal” components.Starting with the V sm,ρ equations, it is clear that for fixed mode m , one can solve theserecursively for all non-minimal components in terms of C m +1 ± m and ρ derivatives thereof.This is a consequence of being in Fefferman-Graham gauge, whereby A ρ = − A ρ = V ,and we will remain in this gauge throughout this paper. Having solved for the minimalcomponents C m +1 ± m , one should view the V sm,z and V sm, ¯ z equations as determining these interms of C and ( z, ¯ z ) derivatives thereof.This reveals a useful strategy for extracting the smallest possible set of equations toobtain the scalar equation in any background. We think of the ρ equations as implicitlysolved. Then, along ( z, ¯ z ), one need only keep track of the mode indices appearing in anygiven equation, and one need only look at equations along minimal directions.Let us demonstrate with the AdS connection (3.8). We use the following heuristicfor which components appear in which equations (structure constants implied): for modes m = 0 , , Upon solving for the C sm in terms of C , one observes the following pole structure in the λ plane: C sm ∼ ( · · · ) s − Y p =1 ( λ − p ) − (3 . p = 1, as hs[ λ ] degeneratesto SL(N) at integer values of λ ≥
2, and the spin s > λ fields do not exist. The singularity at λ = 1 has a different role, but is a reflection of the fact that hs[1], in the absence of a rescaling ofgenerators, becomes similarly degenerate as many structure constants vanish. V , ¯ z ∼ ∂C + C , V ,z ∼ ∂C + C − V , ¯ z ∼ ∂C + C , V ,z ∼ ∂C + C V , ¯ z ∼ ∂C + C , V ,z ∼ ∂C + C ... ... (3 . V , ¯ z , V ,z form a closed set among components with m = 0 ,
1, and so willbe enough, along with whatever ρ equations we need, to find the Klein-Gordon equation.This is exactly what we presented in (3.11). To warm up to the higher spin connections we will ultimately consider, we present thesimplest possible higher spin deformation of AdS: a constant, chiral spin-3 deformation, A = e ρ V dz − ηe ρ V d ¯ z + V dρA = e ρ V − d ¯ z − V dρ (3 . η .Using our mnemonic of the previous subsection, we make quick work of this connection.Again writing the C master field equation in spacetime and gauge components, we showsome of the ¯ z equations: V − , ¯ z ∼ ∂C − + C + ηC − V , ¯ z ∼ ∂C + C + ηC − V , ¯ z ∼ ∂C + C + ηC − V , ¯ z ∼ ∂C + C + ηC (3 . z are unchanged from the AdS case. Reinserting the spin indices,the minimal closed set of equations is { V ,z , V ,z , V ,z , V , ¯ z } , along with any ρ equationsnecessary. Solving this system gives the following equation for C : (cid:2) ∂ ρ + 2 ∂ ρ + 4 e − ρ ( ∂∂ − η∂ ) − ( λ − (cid:3) C = 0 (3 . s deformation: for a connection A = e ρ V dz − ηe ( s − ρ V ss − d ¯ z + V dρA = e ρ V − d ¯ z − V dρ (3 . (cid:2) ∂ ρ + 2 ∂ ρ + 4 e − ρ ( ∂∂ + η ( − ∂ ) s ) − ( λ − (cid:3) C = 0 (3 . Starting from (3.15), we wish to allow η to have arbitrary dependence on ( z, ¯ z ). This will act as a source for spin-3 charge J (3) , enabling us to compute the correlator hO ( z ) O ( z ) J (3) ( z ) i , where O is a scalar primary dual to C . To do so, we will need toobtain the scalar equation to linear order in the spin-3 source (which we now label µ (3) ).Previous work on higher spin gravity [32,27], following earlier work in pure gravity [33],laid out a dictionary for relating sources and charges to components of the Chern-Simonsconnection, and we will apply and recapitulate those techniques here.The following is a flat connection, to linear order in the source µ (3) ( z ): A z = e ρ V + 1 B (3) J (3) ( z ) e − ρ V − A ¯ z = − X n =0 n ! (( − ∂ ) n µ (3) ( z )) e (2 − n ) ρ V − n A ¯ z = e ρ V − (3 . A ρ = − A ρ = V , subject to ∂J (3) ( z ) = − B (3) ∂ µ (3) ( z ) (3 . λ ] insteadof SL(3,R) and with vanishing stress tensor. The leading term in A ¯ z , namely − µ (3) e ρ V ,is the source, dual to the charge term B (3) J (3) e − ρ V − in A z . The remaining terms in A ¯ z are required for flatness. We have included a normalization constant B (3) in the definitionof the current. While its actual value is unimportant for the calculations in this paper ,we include it to stress that we are inserting the factor e R d z µ (3) J (3) (3 . When writing the functional dependence of fields and operators on ( z, ¯ z ), we temporarily usethe notation z ≡ ( z, ¯ z ). For an explicit formula for this coefficient for any spin, see [21], equation (A.4), where B ( s ) = − k π N s in their notation.
13n the CFT path integral. Writing the connection with unit coefficient (up to a sign) forthe chemical potential µ (3) then fixes the other free coefficient, and B (3) is fixed by theWard identity (3.21), equivalently by the OPE J (3) ( z ) J (3) (0) ∼ B (3) π z (3 . ϕ µνσ ∼ Tr( e ( µ e ν e γ ) ),one can explicitly check that this connection turns on various components of ϕ µνσ : forinstance, the component ϕ ¯ z ¯ z ¯ z ∼ µ (3) e ρ Tr( V V − V − ) (3 . C : V ,z , V ,z , V − ,z , V − ,z , V , ¯ z , V , ¯ z , V − , ¯ z (3 . V sm,ρ equations required toeliminate non-minimal components of C . Solving these perturbatively, one finds the fol-lowing scalar equation to linear order in µ (3) ≡ µ :( (cid:3) KG + (cid:3) µ ) C = 0 (3 . (cid:3) KG = ∂ ρ + 2 ∂ ρ + 4 e − ρ ∂∂ − ( λ − (cid:3) µ = − e − ρ (cid:16) ∂ µ∂ + ∂ µ∂∂ (cid:17) + 4 B (3) J (3) e − ρ ∂ + 13 e − ρ (cid:0) ∂ µ∂ − ∂ µ∂∂ ρ + ∂ µ∂∂ − ∂ µ∂∂∂ ρ (cid:1) − e − ρ (cid:16) ∂ µ∂ ρ − ( λ − ∂ µ + 3 ∂ µ∂∂ ρ − ∂ µ∂∂ ρ − ( λ − ∂ µ∂ + 36 ∂µ∂ − ∂µ∂ ∂ ρ + 24 µ∂ (cid:17) (3 . C ( ρ, z ; z ) = G b∂ ( ρ, z ; z ) + φ µ ( ρ, z ; z ) (3 . µ . The first term is the bulk-to-boundary prop-agator of a scalar in AdS with m = λ −
1, and with standard (+) or alternative ( − )quantization: G b∂ ( ρ, z ; z ) = ± λπ (cid:18) e − ρ e − ρ + | z − z | (cid:19) ± λ (3 . µ is φ µ ( ρ, z ; z ) = − Z d z ′ dρ ′ e ρ ′ G bb ( ρ, z ; ρ ′ , z ′ ) (cid:3) ′ µ G b∂ ( ρ ′ , z ′ ; z ) (3 . G bb ( ρ, z ; ρ ′ , z ′ ) is the bulk-to-bulk propagator obeying (cid:3) KG G bb ( ρ, z ; ρ ′ , z ′ ) = e − ρ δ ( ρ − ρ ′ ) δ (2) ( z − z ′ ) (3 .
4. Three-point correlators from the bulk
We now turn to the main focus of this paper: the efficient computation of three-pointcorrelation functions involving two scalar operators and one higher spin current. Ourbasic observation is that starting from the solution for a free scalar field in AdS we cangenerate a new solution by performing a higher spin gauge transformation. This essentiallyreduces the whole problem to determining how the scalar field transforms under the gaugetransformation.
To illustrate our general approach in the simplest context, in this section we computethe three-point function of two scalar operators and a spin-1 current. Rather than workingin the Vasiliev theory, here we take the bulk action to be a complex scalar field of mass m = λ − S = k π Z A ∧ dA + 12 Z d x √ g (cid:16) | D µ φ | + ( λ − | φ | (cid:17) (4 . D µ = ∂ µ + A µ . To compute the correlator hO ( z ) O ( z ) J (1) ( z ) i we proceed as follows.We insert delta function sources at z and z by imposing the following asymptotic behavioron the scalar and gauge field b φ ( ρ, z ) ∼ µ φ δ (2) ( z − z ) e − (1 − λ ) ρ , b A z ( ρ, z ) ∼ µ A δ (2) ( z − z ) , ρ → ∞ (4 . The reason for the hats will be clear momentarily. Also, note here that we are using standardquantization for the scalar. µ φ µ A contribution to the vev O ( z ), whichalso can be read off from the scalar field asymptotics, b φ ( ρ, z ) ∼ O ( z ) B φ e − (1+ λ ) φ , ρ → ∞ , z = z , (4 . B φ unspecified, though we note that a consistent holographicdictionary fixes B φ = 2 λ (for λ = 0) [37]. The three-point function is then given by O ( z ) = µ φ µ A hO ( z ) O ( z z ) J (1) ( z ) i + · · · (4 . · · · denote terms of other order in µ φ,A .Since the gauge field has no propagating degrees of freedom we can generate therequired solution by a gauge transformation. In particular, we start by solving for thescalar field with A = 0, using the bulk-boundary propagator φ ( ρ, z ) = Z d z ′ G b∂ ( ρ, z ; z ) φ − ( z ′ ) (4 . φ − ( z ). To generate the desired gaugefield solution we apply a gauge transformation A µ = ∂ µ Λ , Λ( z ) = µ A π z − z (4 . ∂ z (cid:0) z (cid:1) = 2 πδ (2) ( z ). The gauge transformation acts on the scalarfield as φ ( ρ, z ) → b φ ( ρ, z ) = (cid:0) − Λ( z ) (cid:1) φ ( ρ, z )= (cid:0) − Λ( z ) (cid:1) Z d z ′ G b∂ ( ρ, z ; z ) φ − ( z ′ ) (4 . b φ ( ρ, z ) ∼ (cid:0) − Λ( z ) (cid:1) φ − ( z ) e − (1 − λ ) ρ , ρ → ∞ (4 . b φ − ( z ) = (cid:0) − Λ( z ) (cid:1) φ − ( z ) (4 . b φ − ( z ) = µ φ δ (2) ( z − z ) gives us an expression for theoriginal source φ − ( z ) = µ φ (cid:0) z ) (cid:1) δ (2) ( z − z ) (4 . z = z , .Retaining just the term of order µ φ µ A , we find b φ ( ρ, z ) ∼ λπ µ φ (cid:18) Λ( z ) | z − z | λ ) − Λ( z ) | z − z | λ ) (cid:19) e − (1+ λ ) ρ , ρ → ∞ (4 . O ( z ) = λµ φ B φ π Λ( z ) − Λ( z ) | z | λ ) ! (4 . hO ( z ) O ( z ) J (1) ( z ) i = λB φ π (cid:18) z z z (cid:19) | z | λ ) = 12 π (cid:18) z z z (cid:19) hO ( z ) O ( z ) i (4 . , as given by the bulk to boundary propagator, all we need to dois to perform a gauge transformation to generate a solution with the required asymptoticbehavior of the gauge field. From this solution we read off the scalar vev, and thence thethree-point function. We now establish an algorithm for computing hO ± ( z ) O ± ( z ) J ( s ) ( z ) i for arbitrary s , where we take O ± and its complex conjugate to be dual to the complex scalar field C in either standard (+) or alternate ( − ) quantization. We will work in the standardquantization throughout and include the alternate quantization at the end by taking λ →− λ . In addition, the case of the second complex bulk scalar e C , dual to an operator e O ± and its complex conjugate, will be read off afterwards.Our starting point is the equation (3.1) which we reproduce here: dC + A ⋆ C − C ⋆ A = 0 (4 . λ ] ⊕ hs[ λ ] gauge invariance A → A + d Λ + [ A, Λ] ⋆ A → A + d Λ + [ A, Λ] ⋆ C → C + C ⋆ Λ − Λ ⋆ C (4 . s source in the unbarred sector by performinga chiral gauge transformation with parameterΛ( ρ, z ) = s − X n =1 n − − ∂ ) n − Λ ( s ) ( z ) e ( s − n ) ρ V ss − n (4 . A ¯ z , δA z = ∂ z Λ ( s ) e ( s − ρ V ss − + · · · (4 . A z , δA z = 1(2 s − ∂ s − Λ ( s ) e − ( s − ρ V s − ( s − (4 . C field using (4.15), which wedenote with a hat, gives b C = C + ( δC ) = C − (Λ ⋆ C ) (4 . δC ) :( δC ) = − s − X n =1 n − − ∂ ) n − Λ ( s ) · g ss s − ( s − n, n − s ) C s − ( s − n ) e ( s − n ) ρ (4 . C s − ( s − n ) of the master field C in AdS. Asdiscussed previously, these are all fixed on-shell in terms of C and its derivatives. Oncewe write ( δC ) in terms of the AdS scalar C to obtain an expression analogous to (4.7),the remaining work follows the spin-1 example of subsection 4.1.To organize the calculation, we first derive a general formula for the correlator, de-laying presentation of the explicit formulae for C s − ( s − n ) to the next subsection. We canrewrite (4.20) as ( δC ) = D ( s ) C (4 . s -dependent differential operator D ( s ) which contains derivatives ( ∂, ∂ ρ ). Substi-tution for the C s − ( s − n ) will reveal that in the sum (4.20), only the terms for which n ≤ s will be needed for our computation: these have no ρ -dependence, while the n > s termsdecay at the AdS boundary. This will imply that D ( s ) is of order s − ∂ , sowe can decompose D ( s ) as D ( s ) = s X n =1 f s,n ( λ, ∂ ρ ) ∂ n − Λ ( s ) ∂ s − n (4 . f s,n ( λ, ∂ ρ ).We now switch to the notation C ≡ φ (4 . See equations 4.40, 4.42. b φ ( ρ, z ) ∼ (1 + D ( s ) ) e − (1 − λ ) ρ φ − ( z ) , ρ → ∞ (4 . D ( s ) through the ρ -dependent prefactor, we define D ( s ) ± ≡ D ( s ) ( ∂ ρ → − (1 ± λ )) (4 . f s,n ± ( λ ). Then setting the transformed source b φ − ( z ) = µ φ δ (2) ( z − z ) andinverting to linear order, φ − ( z ) = µ φ (1 − D ( s ) − ) δ (2) ( z − z ) (4 . z = z now reads, omitting theleading part that is local in the higher spin source, b φ + ( z ) = λµ φ π Z d z ′ (1 + D ( s )+ ( z )) (1 − D ( s ) − ( z ′ )) δ (2) ( z ′ − z ) | z − z ′ | λ ) , ρ → ∞ (4 . ∂ → ∂ z . Isolating the piece of order µ φ D ( s ) andplacing our scalar operator at the boundary point z = z , we have b φ + ( z ) = λµ φ π " D ( s )+ ( z ) · | z | λ ) − Z d z ′ D ( s ) − ( z ′ ) δ (2) ( z ′ − z ) | z − z ′ | λ ) (4 . Z d z ′ D − ( z ′ ) δ (2) ( z ′ − z ) | z − z ′ | λ ) = s X n =1 f s,n − ( λ ) Z d z ′ ∂ n − z ′ a∂ s − nz ′ δ (2) ( z ′ − z ) | z − z ′ | λ ) (4 . n ’th term is( − s − n f s,n − ( λ ) ∂ s − nz ∂ n − z Λ ( s ) | z | λ ) ! =( − s − n f s,n − ( λ ) s − n X j =0 (cid:18) s − nj (cid:19) [ ∂ n − jz Λ ( s ) ] ∂ s − n − jz | z | λ ) (4 . ∂ nz ′ | z − z ′ | λ ) = ( − n ∂ nz | z − z ′ | λ ) = Γ( λ + n + 1)Γ( λ + 1) 1( z − z ′ ) n | z − z ′ | λ ) (4 . f s,n ± ( λ ) and the transfor-mation parameter Λ ( s ) : b φ + ( z ) = λµ φ π | z | λ ) " s X n =1 (cid:18) − z (cid:19) s − n n f s,n + ( λ ) Γ( λ + s − n + 1)Γ( λ + 1) ∂ n − z Λ ( s ) − f s,n − ( λ ) s − n X j =0 (cid:18) s − nj (cid:19) Γ( λ + s − n − j + 1)Γ( λ + 1) ( ∂ n − jz Λ ( s ) ) z j o (4 . z , we take (cf. (4.17))Λ ( s ) = 12 π z − z (4 . b φ + ( z ) = ( − s − λµ φ π | z | λ ) s X n =1 z s − n ( f s,n + ( λ ) Γ( λ + s − n + 1)Γ( λ + 1) ( n − z n − f s,n − ( λ ) 1 z n s − n X j =0 ( − j (cid:18) s − nj (cid:19) Γ( λ + s − n − j + 1)Γ( λ + 1) ( n − j )! (cid:18) z z (cid:19) j ) (4 . hO + ( z ) O + ( z ) J ( s ) ( z ) i boils down toknowing the functions f s,n ± ( λ ) encoding the change in the scalar under gauge transforma-tion.Before turning to the problem of determining these functions f s,n ± ( λ ), we note that aconformally symmetric result has the property hO ( z ) O ( z ) J ( s ) ( z ) i = ( − s hO ( z ) O ( z ) J ( s ) ( z ) i (4 . b φ + ( z ) = λµ φ π h D ( s )+ ( z ) + ( − s D ( s )+ ( z ) i · | z | λ ) (4 . f s,n ± ( λ ) this looks like b φ + ( z ) = ( − s − λµ φ π | z | λ ) s X n =1 f s,n + ( λ ) z s − n Γ( λ + s − n + 1)Γ( λ + 1) ( n − (cid:18) z n + ( − n z n (cid:19) (4 . − n comes from sign changes under derivatives acting on | z | (cf.(4.31)). One can show, by using (4.34) and (4.37) and equating terms with the samenumber of powers of z , that this in turn implies the following unobvious relations: f s,j + ( λ ) = − s X n =1 ( − n f s,n − ( λ ) (cid:18) s − nj − n (cid:19) (4 . j .Upon solving for the f s,n ± ( λ ) generated by the gauge transformation (4.16), we willshow that (4.38) is indeed satisfied. C To write the f s,n ± ( λ ) as defined by (4.20), (4.21) and (4.22), we need a formula for allcomponents of the master field C in AdS, written in terms of C ≡ φ . This amounts tosolving (3.10). We first solve for the minimal components C m +1 ± m in terms of φ , then forthe non-minimal components C s = m +1 ± m in terms of the C m +1 ± m , and finally we put the twotogether. Minimal components C m +1 ± m : Taking m → − m and s = m + 1 in the second equation in (3.10) gives ∂ z C m +1 − m + e ρ g m +2)3 (1 , − m − C m +2 − m − = 0 (4 . C sn one needs | n | ≤ s −
1. Solving recursivelyyields the following expression: C m +1 − m = m +1 Y p =2 g p (1 , − p ) ! − ( − e − ρ ∂ z ) m φ (4 . z yields C m +1 m = m +1 Y p =2 g p ( − , p − ! − (2 e − ρ ∂ ¯ z ) m φ (4 . g st ( m, n ) = g ts ( n, m ). Non-minimal components C s = m +1 ± m : From the ρ equations in (3.10), one can see that, say, the component C will have thesame structure, when written in terms of φ , as C in terms of C , and so on. In general,components with fixed s − | m | when expressed as a function of their respective minimalcomponents will have the same form in terms of structure constants.21he solution is C s ± m = ( − s − − m s Y p =2+ m g p ( m, ! − ⌊ s − − m ⌋ X α =0 A α ( s, m ) ∂ s − α − m − ρ C m +1 ± m (4 . A α ( s, m ) are defined as A α ( s, m ) = ( − α X i ...i α α Y k =1 g i k ( m,
0) (4 . k + m ≤ i k ≤ k + s − − αi k ≥ i k − + 2 , ∀ k ≥ . C s obtained from solving (3.10)directly to facilitate easy comparison with (4.42).Via (4.40), (4.41) and (4.42), one has all components of C in terms of the fundamentalscalar field, φ . These formulae also justify our previous statements about subleading terms,and about D ( s ) being order s − ∂ .With these results in hand, we combine (4.40) and (4.42) at m = s − n to write C s − ( s − n ) in terms of φ , and plug into (4.20). The result can be written( δC ) = s X n =1 f s,n ± ( λ ) ∂ n − z Λ ( s ) ∂ s − nz φ + (subleading)= s X n =1 ( − s s − n − ( n − F ± ( s, s − n ; λ ) ∂ n − z Λ ( s ) ∂ s − nz φ (4 . F ± ( s, s − n ; λ ) = g ss s − ( s − n, n − s ) × s Y p =2+ s − n g p ( s − n, ! − s − n +1 Y p =2 g p (1 − p, ! − × ⌊ n − ⌋ X α =0 A α ( s, s − n )( ± λ + s − n + 1) n − − α (4 . A α ( s, s − n ) defined as in (4.43) and (4.44). We have dropped the subleading terms inthe second line of (4.45). To obtain this result we have used the replacement ∂ ρ → − (1 ± λ )and the identity n − − α X χ =0 (cid:18) n − − αχ (cid:19) ( s − n ) n − − α − χ (1 ± λ ) χ = ( ± λ + s − n + 1) n − − α (4 . f s,n ± ( λ ) : f s,n ± ( λ ) = ( − s s − n − ( n − F ± ( s, s − n ; λ ) (4 . f s,n ± ( λ ) when the structureconstants are written out explicitly. We set out to compute the f s,n ± ( λ ) for low values of n , using the formulae (4.46),(4.48). The results up to s = 8 are: f s, ± ( λ ) = ( − s f s, ± ( λ ) = ( − s s ± λ )Γ( s − ± λ ) f s, ± ( λ ) = ( − s s − s − s ± λ )Γ( s − ± λ ) f s, ± ( λ ) = ( − s s − s − s ± λ )Γ( s − ± λ ) f s, ± ( λ ) = ( − s
96 ( s − s − s − s −
5) Γ( s ± λ )Γ( s − ± λ ) f s, ± ( λ ) = ( − s
960 ( s − s − s − s −
5) Γ( s ± λ )Γ( s − ± λ ) f s, ± ( λ ) = ( − s s − s − s − s − s − s −
7) Γ( s ± λ )Γ( s − ± λ ) f s, ± ( λ ) = ( − s s − s − s − s − s − s −
7) Γ( s ± λ )Γ( s − ± λ ) (4 . f s,n ± ( λ ) = ( − s Γ( s ± λ )Γ( s − n + 1 ± λ ) 12 n − (2 ⌊ n ⌋ − ⌊ n − ⌋ ! ⌊ n − ⌋ Y j =1 s + j − n s − j − . → − λ then yields the final answer for the correlator: hO ± ( z ) O ± ( z ) J ( s ) ( z ) i = ( − s − B φ π | z | ± λ ) Γ( s ) Γ(2 s −
1) Γ( s ± λ )Γ( ± λ ) (cid:18) z z z (cid:19) s = ( − s − π Γ( s ) Γ(2 s −
1) Γ( s ± λ )Γ(1 ± λ ) (cid:18) z z z (cid:19) s hO ± ( z ) O ± ( z ) i (4 . n = 13; assuming (4.50), it is simple to confirm (4.51) at any desired spin.Recalling the discussion at the end of section 2, there exists a second projection of themaster field C that gives rise to the equation d e C + A ⋆ e C − e C ⋆ A = 0 (4 . e C is also a complex scalar field, dual to a scalar CFT operator e O and its complex conjugate. This equation is simply related to (4.14) by the exchange A ↔ A , which flips the sign of all odd spin tensors. Therefore, the result for the tildedcorrelator is simply our result (4.51) with a ( − s removed: h e O ± ( z ) e O ± ( z ) J ( s ) ( z ) i = − π Γ( s ) Γ(2 s −
1) Γ( s ± λ )Γ(1 ± λ ) (cid:18) z z z (cid:19) s h e O ± ( z ) e O ± ( z ) i (4 . W N coset CFT duality conjecture, one scalar is in standardquantization and the other in alternate quantization. The operation of flipping the sign ofodd spin fields corresponds to a charge conjugation operation in the CFT.These results match and extend the bulk calculations of [24] which were restricted to λ = 1 /
2. To further compare to the CFT, we now compute the same correlators usingCFT considerations, again for all λ and s , and find perfect agreement with the bulk.
5. Three-point correlators from CFT
We now shift focus and consider the constraints on three-point functions due to theexistence of a higher-spin current algebra. Our considerations will be entirely based onsymmetry, and in particular on the existence of W ∞ [ λ ] current algebra. If these currentsare not present in the CFT, then even before considering scalar operators there will bea mismatch between bulk and boundary correlators involving only currents. So we willassume the existence of this symmetry algebra, and then see what constraints this imposeson the scalar-scalar-current three-point functions. Our computations in this section arealong the same lines as in [21], but generalized to arbitrary s .24uppose we have a spin-s current J ( s ) ( z ), and scalar primary operator O ( z, z ), whoseOPE has the following leading singularity, J ( s ) ( z ) O (0) ∼ A ( s ) z s O (0) + · · · (5 . hO ( z ) O ( z ) J ( s ) ( z ) i = A ( s ) (cid:18) z z z (cid:19) s hO ( z ) O ( z ) i (5 . J ( s ) ( z ) = − π ∞ X m = −∞ W ( s ) m z m + s (5 . |Oi as J ( s )0 |Oi = − πA ( s ) |Oi (5 . J ( s ) ( z ), s = 2 , , . . . , obey the W ∞ [ λ ] currentalgebra. The wedge modes are defined to be those that annihilate the vacuum state, andare given by V sm = J ( s ) m , | m | < s (5 . W ∞ [ λ ], as their com-mutators yield modes outside the wedge; in particular, this is due to the nonlinearitiespresent in W ∞ [ λ ]. However, the nonlinear terms are suppressed at large central charge,and so in the limit c → ∞ the wedge modes do define a subalgebra of W ∞ [ λ ] – the wedgesubalgebra. This subalgebra is hs[ λ ]. To exploit this simplification, for the remainder ofthis section we will assume that we are working in the limit of large central charge. See[21] for further discussion.Furthermore, these considerations fix the relative normalization of the bulk and bound-ary currents. In particular, we have defined our conventions such that the currents (5.3)are equal to the currents derived from the bulk.Acting on |Oi with the wedge modes, we obtain a representation of hs[ λ ], and we cantherefore use the representation theory of hs[ λ ] to determine the zero mode eigenvaluesappearing in (5.4), and thence the three point function (5.2). The Virasoro zero modeeigenvalue is fixed by the scaling dimension of O , V |O ± i = 12 (1 ± λ ) |O ± i (5 . We use the shorthand O ( z, z ) = O ( z ) in what follows. O ± . We need to compute theremaining eigenvalues.To proceed, it is useful to build up the hs[ λ ] generators in terms of SL(2) generators.We write V = L , V = L , V − = L − , which obey the SL(2) algebra[ L , L − ] = 2 L , [ L , L ] = − L , [ L , L − ] = L − (5 . V sm generators as V sm = ( − m − ( m + s − s − (cid:2) L − , . . . [ L − , [ L − , | {z } s − − m L s − ]] (cid:3) (5 . C = L −
12 ( L L − + L − L ) = 14 ( λ −
1) (5 . s = 3 : V |O ± i = − (cid:2) L − , [ L − , L ] (cid:3) = (cid:18) C − L (cid:19) |O ± i = −
16 ( λ ± λ ± |O ± i (5 . V s will be a polynomial in λ of degree s −
1, as illustrated in(5.10). This follows, since the terms in (5.8) obtained after working out all the commutatorswill each have s − L = (1 ± λ ) will convert agenerator into at most one power of λ .Next consider taking λ = N , a positive integer. In this case, after factoring outan ideal, hs[ λ ] becomes SL(N), which we can represent in terms of N × N matrices. Inthis representation the generators V sm with s > N all vanish identically when they areconstructed using (5.8), and in particular this holds for the zero modes V s . We also notethat the eigenvalues of L = V in the N × N matrix representation are: − N − , − N − +1 , . . . , N − . Note that the smallest eigenvalue coincides with (1 − λ ), i.e. with theeigenvalue of L acting on |O − i . Together, these facts imply that V s |O − i = 0 for λ =1 , , . . . , s −
1. Combining this with the statement in the previous paragraph, we fix the λ dependence of the zero-mode eigenvalues to be V s |O − i = N ( s )[ λ − ( s − · · · [ λ − λ − |O − i = N ( s )( − s − Γ( s − λ )Γ(1 − λ ) |O − i (5 . As will be discussed at the end of this section, we obtain a second inequivalent representationby appending a factor of ( − s to the generators. N ( s ). To obtain the eigenvalues for |O + i we simply flip the sign of λ ,and we arrive at V s |O ± i = N ( s )( − s − Γ( s ± λ )Γ(1 ± λ ) |O ± i (5 . N ( s ) we can pick some convenient value of λ . If we take λ = 1 / L = −
14 ( x∂ x + ∂ x x ) , L − = 12 ∂ x , L = 12 x (5 . ] using (5.8).Now, at λ = 1 / L |O + i = |O + i , which in the representation (5.13) isachieved by taking L to act on the function x − . As shown in appendix C the eigenvaluesof the other zero mode generators are then computed to be V s x − = ( − s [( s − (2 s − s − (2 s − x − (5 . O + version of (5.12) at λ = 1 / N ( s ) = − Γ( s ) Γ(2 s −
1) (5 . V s |O ± i = ( − s Γ( s ) Γ(2 s −
1) Γ( s ± λ )Γ(1 ± λ ) |O ± i (5 . hO ± ( z ) O ± ( z ) J ( s ) ( z ) i = ( − s − π Γ( s ) Γ(2 s −
1) Γ( s ± λ )Γ(1 ± λ ) (cid:18) z z z (cid:19) s hO ± ( z ) O ± ( z ) i (5 . λ ] admits the automor-phism V sm → ( − s V sm , which can be thought of as charge conjugation. If we had insteadused the charge conjugate generators in (5.8) then a factor of ( − s would have propa-gated through to the final result (5.17). Equivalently, starting from O ± we can consideroperators e O ± that transform as their charge conjugates. The three-point function of suchoperators is therefore h e O ± ( z ) e O ± ( z ) J ( s ) ( z ) i = − π Γ( s ) Γ(2 s −
1) Γ( s ± λ )Γ(1 ± λ ) (cid:18) z z z (cid:19) s h e O ± ( z ) e O ± ( z ) i (5 . . Discussion The results (5.17)-(5.18) derived from CFT considerations agree perfectly with thecorrelators derived from the higher spin theory in the bulk, namely (4.51) and (4.53).Before discussing the implications of this agreement, let us compare to previous work. Tothis end, we note that in our normalization the current-current two-point function is h J ( s ) ( z ) J ( s ) (0) i = 3 k s − π / sin( πλ ) λ (1 − λ ) Γ( s )Γ( s − λ )Γ( s + λ )Γ( s − ) 1 z s (6 . λ ]algebra with standard normalization.In [24] three-point correlators were computed from the bulk for arbitrary s and λ =1 /
2; and in the ’t Hooft limit of the W N minimal model for s = 3 and arbitrary λ . Theircurrents are normalized to h J ( s ) ( z ) J ( s ) (0) i = z − s . Under the identification O here+ = O CY+ and e O here − = O CY − , and taking into account the different normalizations for the currents,we verify that our results reduce to those of [24] for the special values of s and λ .In [25] the three-point function for s = 4 was computed in the ’t Hooft limit of the W N minimal model. The normalization of the current was not specified, but a normaliza-tion independent ratio was obtained (see equation 3.35 therein). The corresponding ratioobtained from our result is hO + ( z ) O + ( z ) J (4) ( z ) ih e O − ( z ) e O − ( z ) J (4) ( z ) i = (1 + λ )(2 + λ )(3 + λ )(1 − λ )(2 − λ )(3 − λ ) (6 . We now discuss the implications of our agreement between bulk and boundary. Onthe CFT side, what went into the computation was the assumption that the CFT has asymmetry algebra containing hs[ λ ], along with scalar operators of the correct dimension;everything else followed from hs[ λ ] representation theory. So any CFT with these propertieswill have three-point functions that match those of the bulk. One way that hs[ λ ] symmetrycan emerge is if the CFT has W ∞ [ λ ] symmetry, and the central charge is taken to infinity.Then hs[ λ ] is identified with the wedge subalgebra of W ∞ [ λ ].Now consider the case of the W N minimal models proposed by Gaberdiel and Gopaku-mar as CFT duals of the bulk higher spin theory. As we have stressed, even before con-sidering the scalars, it is necessary that in the ’t Hooft limit the CFT acquire W ∞ [ λ ]symmetry if it is to have a chance of matching with the bulk. The bulk theory has such a28ymmetry, and this fixes the form of all correlation functions on the plane involving justcurrents. These will not match with the CFT unless the latter also has W ∞ [ λ ] symmetry.Assuming that the CFT does indeed exhibit hs[ λ ] symmetry, let’s consider the scalars.The results of this paper establish that if the CFT has scalar operators of the correctdimension, ∆ = 1 ± λ , then the scalar-scalar-current three-point functions on the planewill match between the bulk and boundary.As an illustration of these comments, we now establish that a theory of free bosonshas correlators that match those of the corresponding bulk theory. Here by “correlators”we mean those discussed above: namely pure current correlators, and scalar-scalar-currentcorrelators, all evaluated on the plane. Consider the following currents J ( s +2) = − π − s − ( s + 2)!(2 s + 1)!! s X k =0 ( − k s + 1 (cid:18) s + 1 k (cid:19) (cid:18) s + 1 k + 1 (cid:19) ∂ s − k +1 φ∂ k +1 φ (6 . φ is a complex free boson. These currents yield the linear algebra W PRS ∞ [26] at c = 2 [38,39]; the generalization to higher c is obtained by introducing additional copies ofthe free boson. After a nonlinear redefinition of the currents, W PRS ∞ becomes equivalentto W ∞ [1] [40,21]. For λ = 1 the bulk scalar is dual to a CFT operator of dimension 2, andthis is O = ∂φ∂φ . The results of this paper show that the scalar-scalar-current three-pointfunctions of this free boson theory will match those of the higher spin theory in the bulk at λ = 1. For instance, it is simple to check this explicitly for the case of the spin-3 current.A related situation occurs with complex free fermions at λ = 0. The following currents[39] J ( s +2) = 12 π − s − ( s + 1)!(2 s + 1)!! s +1 X k =0 ( − k (cid:18) s + 1 k (cid:19) ∂ s − k +1 ψ∂ k ψ (6 . W ∞ [41] at c = 1. Although W ∞ is not equivalent to W ∞ [0]due to the presence of the spin-1 current in W ∞ , the wedge subalgebra of W ∞ yieldshs[0]; the spin-1 zero mode just yields an operator that commutes with all the other wedgemodes. The scalar operator in this theory is O = ψψ . As we have discussed, this is enoughstructure to guarantee that the scalar-scalar-current correlators (for spin greater than 1)will match those of the bulk theory at λ = 0, and verifying this is straightforward.Note that we are not making any claims here about a full duality between these freeboson/fermion theories and their bulk counterparts. Indeed, without further ingredients itseems clear that the theories cannot be equivalent: if we simply add N copies of the freefields the CFT will have a U ( N ) symmetry along with various nonsinglet operators, noneof which appear to be present in the bulk, at least classically. Acknowledgments
This work was supported in part by the National Science Foundation under Grant No. NSFPHY-07-57702. We are grateful to M. Gutperle for useful conversations, and the KITP for29ospitality. This research was supported in part by the National Science Foundation underGrant No. NSF PHY05-51164. E.P. is supported in part by a UCLA Graduate DivisionDissertation Year Fellowship.
Appendix A. Lightning review of 3D higher spin gravity coupled to scalars
In this appendix we present all details necessary for the bulk theory’s construction,following [20]. In particular we derive the equations (2.11) and (2.19) in section 2.According to [20], the full non-linear system of equations governing the interactionof matter with higher spin gauge fields is formulated in terms of the following generatingfunctions: a spacetime 1-form W = W ν dx ν as well as spacetime 0-forms B and S α . Thegenerating functions
W, B and S α depend on spacetime coordinates x, on auxiliary bosonictwistor variables z α and y α ( α = 1 ,
2) as well as on two pairs of Clifford elements, ψ , and k, ρ : { ψ i , ψ j } = 2 δ ij , kρ = − ρk , k = ρ = 1 (A.1)Moreover, ψ , commute with all other auxiliary variables, and k, ρ obey ky α = − y α k, kz α = − z α kρy α = y α ρ, ρz α = z α ρ (A.2)Indices on z α and y α are raised by ǫ αβ and lowered by the rank two antisymmetric tensor ǫ βα , z α = ǫ αβ z β , z α = z β ǫ βα (A.3)with ǫ αβ ǫ βγ = − δ αγ . We follow the convention ǫ = ǫ = 1 . Using these properties of the auxiliary variabls the basic fields W ν , B and S α can beexpanded in the form A ( z, y ; ψ , , k, ρ | x ) = X b,c,d,e =0 ∞ X m,n =0 m ! n ! A α ...α m β ...β n bcde ( x ) k b ρ c ψ d ψ e z α . . . z α m y β . . . y β n (A.4)where A is either W ν , B or S α . The expression A α ...α m β ...β n bcde ( x ) in equation (A.4) is anordinary spacetime function. Note that A α ...α m β ...β n bcde ( x ) can be choosen to be symmetricin the indices ( α . . . α m ) and in the indices ( β . . . β n ) . In order to formulate the equations of motion we use the Moyal ⋆ -product which actson the twistors y and z in the following way f ( z, y ) ⋆ g ( z, y ) = 1(2 π ) Z d u Z d v e i ( uv ) f ( z + u, y + u ) g ( z − v, y + v ) (A.5)where uv is a short-hand notation, uv = u α v α . We can verify the following commutationrelations: [ y α , y β ] ⋆ = − [ z α , z β ] ⋆ = 2 iǫ αβ , [ y α , z β ] ⋆ = 0 (A.6)30here [ a, b ] ⋆ ≡ a ⋆ b − b ⋆ a is the commutator with respect to the ⋆ -product.In terms of the generating functions W = W ν dx ν , B, S α we are now ready to writedown the full non-linear equations of motion [20]: dW = W ∧ ⋆WdB = W ⋆ B − B ⋆ WdS α = W ⋆ S α − S α ⋆ WS α ⋆ S α = − i (1 + B ⋆ K ) S α ⋆ B = B ⋆ S α (A.7)Here, K – the so-called Kleinian – is given by K = ke i ( zy ) (A.8)These equations (A.7) are invariant under the infinitesimal higher spin gauge transforma-tion δW = dǫ + ǫ ⋆ W − W ⋆ ǫδB = ǫ ⋆ B − B ⋆ ǫδS α = ǫ ⋆ S α − S α ⋆ ǫ (A.9)where ǫ is the infinitesimal gauge parameter which does not depend on ρ, i.e. ǫ = ǫ ( z, y ; ψ , , k | x ) (A.10)We will see that W is the generating function for higher spin gauge fields whereas B isthe generating function for the matter fields. S α will describe auxiliary degrees of freedom.Since the equations of the motion (A.7) possess the symmetry ρ → − ρ and S α → − S α we can truncate the system to the so-called “reduced” system, in which W ν and B areindependent of ρ, while S α is linear in ρ. In this paper we consider the reduced system.
A.1. Vacuum solutions
Here we consider vacuum solutions of the equations of motion (A.7). The fields
B, W and S α of the vacuum solution are denoted by B (0) , W (0) and S (0) α , respectively. In par-ticular we take B (0) to be constant, i.e. B (0) ≡ ν (A.11)Plugging this ansatz into (A.7) we obtain the following three equations dW (0) = W (0) ⋆ ∧ W (0) dS (0) α = W (0) ⋆ S (0) α − S (0) α ⋆ W (0) S (0) α S (0) α = − i (1 + νK ) (A.12)31ote that the other two equations of (A.7) are automatically satisfied by the ansatz (A.11).First, let us discuss the third equation of (A.12). We already mentioned that S α andtherefore also S (0) α is linear in ρ. For the case ν = 0 we can choose S (0) α = ρz α , cf. (A.6).For general ν, S (0) α can be given by S (0) α = ρ ˜ z α (A.13)where we have introduced new auxiliary twistor variables ˜ z α and ˜ y α which are also knownas “deformed oscillators,” ˜ z α = z α + ν w α k ˜ y α = y α + ν w α ⋆ Kw α = ( z α + y α ) Z dt te it ( zy ) (A.14)The deformed oscillators ˜ y α and ˜ z α satisfy the commutation relations[˜ y α , ˜ y β ] ⋆ = 2 iǫ αβ (1 + νk )[ ρ ˜ z α , ρ ˜ z β ] ⋆ = − iǫ αβ (1 + νK )[ ρ ˜ z α , ˜ y β ] ⋆ = 0 (A.15)and therefore it is straightforward to verify that S (0) α , given by equation (A.13), indeedsatisfies the third equation of (A.12).As dS (0) α = 0 , the second equation of (A.12) implies that W (0) should commute with S (0) α , which according to the third line of (A.15) can be achieved by taking W (0) to beindependent of k and ˜ z α . Therefore W (0) is only a function of x, ψ , and ˜ y and can beexpanded as W (0) (˜ y ; ψ , | x ) = X d,e =0 ∞ X n =0 n ! W β ...β n de ( x ) ψ d ψ e ˜ y β ⋆ . . . ⋆ ˜ y β n (A.16)It turns out that we only have to consider symmetric products in ˜ y β ⋆ . . . ⋆ ˜ y β n . Moreover,here we will consider only products with an even number of ˜ y. Under the star productthe auxiliary variables ˜ y α generate the three-dimensional higher spin algebra hs[ λ ]. To bemore precise, a symmetric product ˜ y β ⋆ . . . ⋆ ˜ y β n corresponds to a generator of hs[ λ ] withspin n + 1 . In particular the generators T αβ of the SL(2) subalgebra are given by T αβ = − i { ˜ y α , ˜ y β } ⋆ (A.17)Multiplying symmetrized even-degree polynomials in ˜ y, using the commutation relations asgiven in the first line of equation (A.15) and finally projecting on k = ∓ , the commutation32elation are those of hs[ λ ] with λ = (1 ± ν ) . More details, including the hs[ λ ] the structureconstants, can be found in appendix B.Finally, let us consider the last equation of motion which we have to solve: dW (0) = W (0) ⋆ ∧ W (0) . (A.18)This equation can be written as a flatness condition of a Chern-Simons theory with gaugegroup hs[ λ ] ⊕ hs[ λ ] . In order to see this we introduce hs[ λ ]-valued gauge fields A and A by W (0) = −P + A − P − A (A.19)where A and A are functions of x and ˜ y. We have introduced projection operators P ± = 1 ± ψ P ± ψ = ψ P ± = ±P ± , P ± ψ = ψ P ∓ (A.21)Using (A.19) we can rewrite (A.18) in the following form dA + A ∧ ⋆A = 0 , dA + A ∧ ⋆A = 0 (A.22)which is precisely equation (2.11). Therefore the equations of motion for W (0) are equiv-alent to flatness conditions of gauge fields A and A defined by equation (A.19) . A.2. Matter equations
Let us now linearize the equations of motion (A.7) around the vacuum solution con-structed in the last section. In particular, in this paper we are interested in fluctuations ofthe field B around the constant background B (0) = ν. The fluctuations of B are denotedby C , i.e. B = ν + C (A.23)For W and S α we do not consider any fluctuations. Substituting this ansatz (A.23) intothe equations of motion (A.7) and using the equations (A.12), we obtain two non-trivialequations for C d C − W (0) ⋆ C + C ⋆ W (0) = 0 h S (0) α , C i ⋆ = 0 (A.24)Since S (0) α is given by equation (A.13) we can satisfy the second line of equation (A.24)by demanding that C does not depend on ˜ z α nor on k. Therefore C is only a function of˜ y α , ψ , and x. Typically, C is decomposed as C (˜ y ; ψ , | x ) = C aux (˜ y ; ψ | x ) + C dyn (˜ y ; ψ | x ) ψ (A.25)33t turns out [20] that C aux gives rise to an auxiliary set of fields that can be set to zeroconsistently. By abuse of notation, we will use C instead of C dyn to simplify the notation.Moreover we will decompose C under the projection operators P ± as given in equation(A.20) C = C (˜ y | x ) ψ + e C (˜ y | x ) ψ (A.26)where C = P + C and e C = P − C . If we also express W (0) in terms of gauge fields, eq. (A.19),we can rewrite the second line of (A.24) in the form dC + A ⋆ C − C ⋆ A = 0 d e C + A ⋆ e C − e C ⋆ A = 0 (A.27)These are the equations of linearized matter interacting with an arbitrary higher spinbackground.Comparing the first and second line of equation (A.27) we see that the equations ofmotion for C and e C are related to each other by exchanging A and A. Note that A and A can be written in terms of the generalized vielbein e and generalized spin connection ω,A = ω + e A = ω − e (A.28)Under the exchange of A and A the generalized vielbein e is odd. Therefore the sign ofthe generalized vielbein and hence the sign of any metric-like tensor field of odd spin isflipped under this operation. Appendix B. hs [ λ ] , and Moyal vs. lone-star products The hs[ λ ] structure constants are g stu ( m, n ; λ ) = q u − u − φ stu ( λ ) N stu ( m, n ) (B.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) (B.2)We make use of the descending Pochhammer symbol,[ a ] n = a ( a − ... ( a − n + 1) (B.3)34 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 / y α , ˜ y β ] ⋆ = 2 iǫ αβ (1 + νk ) (B.4)which in our conventions ǫ = ǫ = 1 is[˜ y , ˜ y ] ⋆ = 2 i (1 + νk ) (B.5)The beauty of the deformed oscillators is that under the action of the Moyal product ,their star commutator gives the oscillator algebra (B.4).To compute the Moyal product of two symmetric, even-degree oscillator polynomials,one uses (B.4) to symmetrize the product. To compute the lone-star product of twohs[ λ ] generators, one plugs into formula (3.3). The purpose of this section is to provideevidence that these two multiplications are isomorphic upon identifying the map betweenthe generator and polynomial bases, and using the relation λ = 1 − νk k = 1 is the Clifford element defined in section 2 and in appendix A. To ourknowledge, this has not been proven in the literature.Let us begin with the SL(2) subalgebra spanned by symmetric polynomials S αβ = ˜ y ( α ˜ y β ) (B.7)These obey commutation relations [ S , S ] = 8 iS [ S , S ] = 4 iS [ S , S ] = 4 iS (B.8)Comparing with the SL(2) subalgebra of hs[ λ ] canonically normalized as[ V , V − ] = 2 V [ V , V ] = V [ V , V − ] = V − (B.9) In what follows, every product of ˜ y is implicitly a Moyal product. V = (cid:18) − i (cid:19) S , V = (cid:18) − i (cid:19) S , V − = (cid:18) − i (cid:19) S (B.10)Having fixed (B.10) we can compare, on the one hand, the Moyal product of two S αβ ,and on the other, the lone-star product between two of the V m , and so on for higher spins.The work comes in symmetrizing the oscillator products, through tedious but straightfor-ward application of (B.4). We present results through spin-4:˜ y ˜ y = S + i ( νk + 1)˜ y ˜ y ˜ y ˜ y = S + i ( νk + 3) S ˜ y ˜ y ˜ y ˜ y = S + 4 iS + 23 ( νk + 1)( νk − y ˜ y ˜ y ˜ y ˜ y ˜ y = S + i ( νk + 5) S ˜ y ˜ y ˜ y ˜ y ˜ y ˜ y = S + 8 iS + 45 ( νk + 3)( νk − S ˜ y ˜ y ˜ y ˜ y ˜ y ˜ y = S + i ( νk + 9) S + 25 ( νk + 3)( νk − S + 2 i νk + 3)( νk + 1)( νk −
3) (B.11)All remaining products can be found by commutation or taking the adjoint, ˜ y ↔ ˜ y , i →− i . Using these, one can show by explicit computation that, at least through spin-4, anyMoyal product of oscillator polynomials maps to the lone-star product of hs[ λ ] generatorsupon making the identification V sm = (cid:18) − i (cid:19) s − S sm (B.12)where S sm is the symmetrized product of 2 s − m defined as2 m = N − N (B.13)The prefactor depends on the hs[ λ ] normalization factor q = 1 / V ± ) s − = V s ± ( s − (B.14)implies that this is trivially true for elements with m = ± ( s − S ⋆ S = S S ⋆ S = S S ⋆ S = S − iS S ⋆ S = S + 2 iS S ⋆ S = S + 4 iS + 23 ( νk + 1)( νk − S ⋆ S = S −
13 ( νk + 1)( νk −
3) (B.15)We now compare this to the lone-star products of spin-2 generators: V ⋆ V = V V − ⋆ V − = V − V ⋆ V = V − V V ⋆ V − = V − + 12 V − V ⋆ V − = V + V − (cid:18) λ − (cid:19) V ⋆ V = V + λ −
112 (B.16)These are isomorphic under (B.6) and (B.12).A less trivial example is the product S ⋆ S = 14 (˜ y ˜ y ˜ y ˜ y + ˜ y ˜ y ˜ y ˜ y + ˜ y ˜ y ˜ y ˜ y + ˜ y ˜ y ˜ y ˜ y )˜ y ˜ y = S + 6 iS + 25 ( νk + 3)( νk − S (B.17)Compare to the lone-star product V ⋆ V − = V + 32 V − (cid:18) λ − (cid:19) V (B.18)These are isomorphic under (B.6) and (B.12).We conjecture that the isomorphism is valid for all spins under the identification(B.12). 37 ppendix C. Derivation of (5.14) Given L = −
14 ( x∂ x + ∂ x x ) , L − = 12 ∂ x , L = 12 x (C.1)and V sm = ( − m − ( m + s − s − (cid:2) L − , . . . [ L − , [ L − , | {z } s − − m L s − ]] (cid:3) (C.2)we need to show V s x − = ( − s [( s − (2 s − s − (2 s − x − (C.3)We start from e tL − f ( x ) e − tL − = f ( x + t∂ x ) (C.4)which follows by thinking of L − as the Hamiltonian for a free particle. In particular, thisimplies e tL − ( L ) s − e − tL − = 12 s − ( x + t∂ x ) s − (C.5)Expanding the left hand side in powers of t , the desired term yielding V s is the t s − term.This term preserves the power of x , and so we can write V s x − = − (cid:20) [( s − s − (2 s − x + ∂ x ) s − x − (cid:21) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − term (C.6)To extract the x − term on the right hand side we write a contour integral and integrateby parts,( x + ∂ x ) s − x − (cid:12)(cid:12)(cid:12) coeff of x − = 12 πi I dzz ( z + ∂ ) s − z = 12 πi I dz z ( z − ∂ ) s − z (C.7)Now write ( z − ∂ ) s − as the t s − term in e t ( z − ∂ ) = e − t / e tz e − t∂ and perform the integral.This yields ( x + ∂ x ) s − x − (cid:12)(cid:12)(cid:12) coeff . of x − = ( − s − (2 s − Appendix D. Low spin results: C s in AdS We present the explicit formulae for the components C s of the master field in AdS,through s = 8, obtained by recursive solution of the V s ,ρ equations in AdS: ∂ ρ C s + 2 C s − + C s +10 g ( s +1)23 (0 ,
0) = 0 (D.1)38hese can be compared to the spin s formula (4.42). We use the temporary notation g s (0 , ≡ g s (D.2)and the following expressions act on C : C = − ( g ) − ∂ ρ C = ( g g ) − · ( ∂ ρ − g ) C = ( g g g ) − · (cid:0) − ∂ ρ + 2( g + g ) ∂ ρ (cid:1) C = ( g g g g ) − · (cid:0) ∂ ρ − g + g + g ) ∂ ρ + 4 g g (cid:1) C = ( g g g g g ) − · (cid:0) − ∂ ρ + 2( g + g + g + g ) ∂ ρ − g ( g + g ) + g g ) ∂ ρ (cid:1) C = ( g g g g g g ) − · (cid:16) ∂ ρ − g + g + g + g + g ) ∂ ρ + 4( g ( g + g + g ) + g ( g + g ) + g g ) ∂ ρ − g g g (cid:17) C = ( g g g g g g g ) − · (cid:16) − ∂ ρ + 2( g + g + g + g + g + g ) ∂ ρ − g ( g + g + g + g ) + g ( g + g + g ) + g ( g + g ) + g g ) ∂ ρ + 8( g ( g ( g + g ) + g g ) + g g g ) (cid:17) ∂ ρ (D.3)39 eferences [1] P. Haggi-Mani and B. Sundborg, “Free large N supersymmetric Yang-Mills theory asa string theory,” JHEP , 031 (2000) [arXiv:hep-th/0002189].[2] B. Sundborg, “Stringy gravity, interacting tensionless strings and massless higherspins,” Nucl. Phys. Proc. Suppl. , 113 (2001) [arXiv:hep-th/0103247].[3] E. Sezgin and P. Sundell, “Massless higher spins and holography,” Nucl. Phys. B ,303 (2002) [Erratum-ibid. B , 403 (2003)] [arXiv:hep-th/0205131].[4] C. Fronsdal, “Massless Fields With Integer Spin,” Phys. Rev. D , 3624 (1978).[5] E. S. Fradkin and M. A. Vasiliev, “On the Gravitational Interaction of Massless HigherSpin Fields,” Phys. Lett. B , 89 (1987).[6] E. S. Fradkin and M. A. Vasiliev, “Cubic Interaction in Extended Theories of MasslessHigher Spin Fields,” Nucl. Phys. B , 141 (1987).[7] M. A. Vasiliev, “Consistent equation for interacting gauge fields of all spins in (3+1)-dimensions,” Phys. Lett. B , 378 (1990).[8] M. A. Vasiliev, “Closed equations for interacting gauge fields of all spins,” JETP Lett. , 503 (1990) [Pisma Zh. Eksp. Teor. Fiz. , 446 (1990)].[9] M. A. Vasiliev, “More On Equations Of Motion For Interacting Massless Fields OfAll Spins In (3+1)-Dimensions,” Phys. Lett. B , 225 (1992).[10] I. R. Klebanov and A. M. Polyakov, “AdS dual of the critical O(N) vector model,”Phys. Lett. B , 213 (2002) [arXiv:hep-th/0210114].[11] A. C. Petkou, “Evaluating the AdS dual of the critical O(N) vector model,” JHEP , 049 (2003). [hep-th/0302063].[12] R. G. Leigh, A. C. Petkou, “Holography of the N=1 higher spin theory on AdS(4),”JHEP , 011 (2003). [hep-th/0304217].[13] E. Sezgin, P. Sundell, “Holography in 4D (super) higher spin theories and a test viacubic scalar couplings,” JHEP , 044 (2005). [hep-th/0305040].[14] S. Giombi and X. Yin, “Higher Spin Gauge Theory and Holography: The Three-PointFunctions,” JHEP , 115 (2010) [arXiv:0912.3462 [hep-th]].[15] S. Giombi and X. Yin, “Higher Spins in AdS and Twistorial Holography,”arXiv:1004.3736 [hep-th].[16] S. Giombi, X. Yin, “On Higher Spin Gauge Theory and the Critical O(N) Model,”[arXiv:1105.4011 [hep-th]].[17] R. d. M. Koch, A. Jevicki, K. Jin and J. P. Rodrigues, “ AdS /CF T Constructionfrom Collective Fields,” Phys. Rev. D , 025006 (2011) [arXiv:1008.0633].[18] A. Jevicki, K. Jin and Q. Ye, “Collective Dipole Model of AdS/CFT and Higher SpinGravity,” arXiv:1106.3983 [hep-th].[19] M. R. Gaberdiel and R. Gopakumar, “An AdS Dual for Minimal Model CFTs,”arXiv:1011.2986 [hep-th]. 4020] S. F. Prokushkin and M. A. Vasiliev, “Higher spin gauge interactions for massivematter fields in 3-D AdS space-time,” Nucl. Phys. B , 385 (1999) [arXiv:hep-th/9806236].[21] M. R. Gaberdiel and T. Hartman, “Symmetries of Holographic Minimal Models,”arXiv:1101.2910 [hep-th].[22] M. R. Gaberdiel, R. Gopakumar, T. Hartman and S. Raju, “Partition Functions ofHolographic Minimal Models,” arXiv:1106.1897 [hep-th].[23] P. Kraus, E. Perlmutter, “Partition functions of higher spin black holes and their CFTduals,” [arXiv:1108.2567 [hep-th]].[24] C. M. Chang and X. Yin, “Higher Spin Gravity with Matter in AdS and Its CFTDual,” arXiv:1106.2580 [hep-th].[25] C. Ahn, “The Coset Spin-4 Casimir Operator and Its Three-Point Functions withScalars,” [arXiv:1111.0091 [hep-th]].[26] C. N. Pope, L. J. Romans and X. Shen, “W(infinity) and the Racah-Wigner Algebra,”Nucl. Phys. B , 191 (1990).[27] M. Gutperle and P. Kraus, “Higher Spin Black Holes,” JHEP , 022 (2011)[arXiv:1103.4304 [hep-th]].[28] M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter, “Spacetime Geometry in HigherSpin Gravity,” arXiv:1106.4788 [hep-th].[29] A. Castro, E. Hijano, A. Lepage-Jutier and A. Maloney, “Black Holes and SingularityResolution in Higher Spin Gravity,” arXiv:1110.4117 [hep-th].[30] H. S. Tan, “Aspects of Three-dimensional Spin-4 Gravity,” arXiv:1111.2834 [hep-th].[31] M. P. Blencowe, “A Consistent Interacting Massless Higher Spin Field Theory In D= (2+1),” Class. Quant. Grav. , 443 (1989).[32] A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen, “Asymptotic sym-metries of three-dimensional gravity coupled to higher-spin fields,” JHEP , 007(2010) [arXiv:1008.4744 [hep-th]].[33] M. Banados and R. Caro, “Holographic Ward identities: Examples from 2+1 gravity,”JHEP , 036 (2004) [arXiv:hep-th/0411060].[34] J. Hansen, P. Kraus, “Generating charge from diffeomorphisms,” JHEP , 009(2006). [hep-th/0606230].[35] P. Kraus, F. Larsen, “Partition functions and elliptic genera from supergravity,” JHEP , 002 (2007). [hep-th/0607138].[36] P. Kraus, “Lectures on black holes and the AdS(3) / CFT(2) correspondence,” Lect.Notes Phys. , 193-247 (2008). [hep-th/0609074].[37] D. Z. Freedman, S. D. Mathur, A. Matusis, L. Rastelli, “Correlation functions inthe CFT(d) / AdS(d+1) correspondence,” Nucl. Phys. B546 , 96-118 (1999). [hep-th/9804058]. 4138] I. Bakas and E. Kiritsis, “Bosonic realization of a universal W algebra and Z(infinity)Parafermions,” Nucl. Phys. B , 185 (1990) [Erratum-ibid. B , 512 (1991)].[39] E. Bergshoeff, C. N. Pope, L. J. Romans, E. Sezgin and X. Shen, “THE SUPERW(infinity) ALGEBRA,” Phys. Lett. B , 447 (1990).[40] J. M. Figueroa-O’Farrill, J. Mas, E. Ramos, “A One parameter family of Hamiltonianstructures for the KP hierarchy and a continuous deformation of the nonlinear W(KP)algebra,” Commun. Math. Phys. , 17-44 (1993). [hep-th/9207092].[41] C. N. Pope, L. J. Romans and X. Shen, “A NEW HIGHER SPIN ALGEBRA ANDTHE LONE STAR PRODUCT,” Phys. Lett. B242