AdS/CFT for 3D Higher-Spin Gravity Coupled to Matter Fields
aa r X i v : . [ h e p - t h ] N ov EPHOU-13-010November 2013
AdS/CFT for 3D Higher-Spin Gravity Coupled toMatter Fields
Ippei
Fujisawa ∗ , Kenta Nakagawa † and Ryuichi Nakayama ‡ Division of Physics, Graduate School of Science,Hokkaido University, Sapporo 060-0810, Japan
Abstract
New holographic prescription for the model of 3d higher-spin gravity coupledto real matter fields B µν and C , which was introduced in arXiv:1304.7941[hep-th], is formulated. By using a local symmetry, two of the components of B µν are eliminated, and gauge-fixing conditions are imposed such that the non-vanishing component, B φρ , satisfies a covariantly-constancy condition in thebackground of Chern-Simons gauge fields A µ , ¯ A µ . In this model, solutionsto the classical equations of motion for A µ and ¯ A µ are non-flat due to theinteractions with matter fields. The solutions for the gauge fields can, however,be split into two parts, flat gauge fields A µ , ¯ A µ , and those terms that dependon the matter fields. The equations for the matter fields then coincide withcovariantly-constancy equations in the flat backgrounds A µ and ¯ A µ , which areexactly the same as those in linearized 3d Vasiliev gravity. The two- and three-point correlation functions of operators in the boundary CFT are computed byusing an on-shell action, tr ( B φρ C ). This term does not depend on coordinatesdue to the matter equations of motion, and it is not necessary to take the near-boundary limit ρ → ∞ . Analysis is presented for SL(3,R) × SL(3,R) as well as HS [ ] × HS [ ] higher-spin gravity. In the latter model, scalar operators withscaling dimensions ∆ + = and ∆ − = appear in a single quantization. ∗ [email protected] † [email protected] ‡ [email protected] Introduction
Recently, higher-spin gravity theory has been studied extensively. Vasiliev etal proposed non-linear equations of motion for infinite tower of higher-spin gaugefields.[1][2][3][4] Although its description based on an action principle is still underinvestigation, it was conjectured that the higher-spin gravity in 3 dimensions isdual to the 2D W-minimal conformal field theory (CFT) models,[5][6][8][7][9] andthis duality has been studied in the version of the model with linearized scalarfields.[10][11][12][13][14] Asymptotic symmetry algebra of 3d higher-spin gravity isdiscussed in [15], [16].It was also noticed that in 3 dimensions great simplifications occur. The higher-spin fields can be truncated to only those with spin s ≤ N and the theory withnegative cosmological constant in the frame-like approach can be defined in terms ofthe SL ( N, R ) × SL ( N, R ) Chern-Simons (CS) action.[16] Various black hole solutionswere found and their properties were studied.[17][18][21][19][11][20][22] [23][24] Ac-tion integral for massless higher-spin fields in the metric-like approach was proposedby Fronsdal.[25] Correlation functions on the boundary conformal field theory (CFT)are studied by using holographic renormalization.[26] Cubic interaction vertices werealso constructed.[27] Analysis of 3d spin-3 gravity in the metric-like approach wasstudied in [28][29][30].Although 3D higher-spin gauge theory can be formulated in terms of the Chern-Simons (CS) theory, this is just a ‘pure spin-3 gravity’ theory. It is desirable toinclude matter fields. Actually, there are scalar fields in Vasiliev gravity.[3][40] In [30]two of the present authors proposed an action integral for matter fields interactingwith higher-spin gravity. Matter fields are a 0-form C and a 2-form B = B µν dx µ ∧ dx ν . Hamiltonian analysis of this model was performed and it was shown thatfor ‘fixed flat’ CS gauge fields, A, ¯ A = ω ± ℓ e , this model provides Lagrangianformulation of the scalars in linearlized 3d Vasiliev theory.In this paper we will extend the analysis of this model by treating the CS gaugefields as dynamical variables. In this model, the equations for motion of the CS gaugefields, A µ and ¯ A µ , are given by F = − πk CB − (trace term) and ¯ F = − πk BC − (trace term), where F and ¯ F are field strengths, and the gauge fields are generallynot flat even on shell. On the other hand, Vasiliev gravity is a non-Lagrangiantheory, and defined by classical equations of motion; flatness conditions F = ¯ F = 0for gauge fields, and covariantly constancy conditions for scalars, dC + AC − C ¯ A = 02nd d ˜ C + ¯ A ˜ C − ˜ CA = 0.[12] It has been difficult to have an explicit realization of anaction integral which yields these equations of motion. We will analyze the solutionsto the equations of motion for our model, and find that they are also solutions tothe equations of motion for linearized 3d Vasiliev higher-spin gravity with flat gaugefields A µ and ¯ A µ , when the gauge fields A µ and ¯ A µ are split into flat gauge fields A µ and ¯ A µ , and the parts A BCµ and ¯ A BCµ which are written in terms of the matterfields. Hence our model may be regarded as an effective Lagrangian formulation oflinearized 3d Vasiliev gravity. Key point is a local symmetry of our model whichallows us to eliminate two of the three components of the matter field B µν , and toimpose gauge fixing conditions which make the equations of motion for the survivingcomponent B φρ take the same form as those for ˜ C in Vasiliev gravity. This is thefirst main result of this paper.Now that we have a Lagrangian formulation, we will substitute the solutionsto the equations of motion into the action integral, and study AdS/CFT corre-spondence. The correlation functions of the boundary CFT, dual to 3d higher-spingravity coupled with master fields, have been calculated by means of bulk-boundarypropagators.[45][10][11] While this method is very elegant, this does not capturepeculiar properties that stem from the covariant-constancy condition for matterfields, which will be discussed in the following paragraphs. AdS/CFT is a dualitybetween bulk AdS gravity (string) theory and CFT (large N gauge theory) on theboundary.[35][36][37] Explicitly, it states that the generating functional of the cor-relation functions in the boundary CFT is given by the on-shell action of the bulkgravity, with the boundary conditions for the bulk fields on the boundary as thesource functions for the operators. Since there has been no Lagrangian formulationfor 3d higher-spin gravity with matter coupling, this on-shell action method couldnot be employed, and bulk-boundary propagator was used to compute two-pointfunctions. This latter method is based on the fact that in the AdS background thetrace of the scalar field C satisfies massive Klein-Gordon equation.[3][12] Then, inanalogy with the AdS/CFT correspondence for scalar fields in AdS space, two-pointfunctions and the spectrum of conformal weights can be derived from the solutionswith delta function boundary conditions for the scalar fields, by assuming the exis-tence of a conjugate operator. It would, however, be more interesting, if one couldderive the same results by means of the conventional on-shell action method. Fur-thermore, it would be difficult, if not impossible, to generalize the method by means See the last paragraph of sec.3.
3f the bulk-boundary propagator to fields other than scalar fields, such as spin- fields. One of the purposes of this paper is to cope with this problem.We found that the on-shell value of the action integral does not yield a cor-rect generating functional of correlation functions for boundary CFT, which re-spect W algebra symmetry, and at the same time, that this on-shell action canbe eliminated by local boundary counterterms. We then found a new local term,tr ( B φρ ( z, ¯ z, ρ ) C ( z, ¯ z, ρ )), which yields appropriate correlation functions in the bound-ary CFT. Due to the equations of motion (covariantly-constancy conditions), thisterm turns out independent of all coordinates z , ¯ z , ρ . All data on the matter fieldsare stored in the internal space located at a single point in the spacetime. In somesense, holographic screen for the matter fields in this model is inside the internalspace at one point in space time. Hence the near-boundary limit ρ → ∞ is notnecessary. This is the second main result.In our BC model, there are many solutions to the classical equations of motion.Some of them are dual to the primary operators in the boundary CFT, while theothers correspond to the descendants of the primaries. These can be distinguished bycalculating the three-point correlation functions, < O ( z ) ¯ O ( z ) J ( s ) ( z ) > , where O ( z ) and ¯ O ( z ) are operators dual to solutions of the equations of motion, and J ( s ) ( z ), the spin-s current. When O ( z ) and ¯ O ( z ) are primary operators of Walgebra, the three-point functions must satisfy appropriate relations with the two-point functions < O ( z ) ¯ O ( z ) > . The correlation functions of scalar operatorsobtained agree with those obtained by means of the bulk-boundary-propagator. Wealso computed three-point functions of two-primary operators and a spin-s current bycalculating the gauge variation of the on-shell action. Although the action integralis invariant under residual gauge transformation after gauge fixing, the solutions tothe equations of motion are not, and the on-shell action changes.In the usual AdS/CFT correspondence for scalar fields, there exist two prescrip-tions for obtaining correlation functions of CFT; standard and alternate quantiza-tions. [39][38] For some range of the scalar mass, there are two operators O ± withscaling dimensions ∆ ± . Only a single operator can be realized in CFT in each quan-tization. In the HS [ λ ] × HS [ λ ] higher-spin gravity theory coupled to matter fields,there also exist operators O (0 , , O (1 , and their conjugates ¯ O (0 , , ¯ O (1 , with twodifferent scaling dimensions in the boundary CFT. It is shown that in our prescrip-tion for computing correlation functions with the boundary action tr ( B φρ C ), bothtypes of operators can be quantized at the same time, and the off-diagonal correla-4ors such as h O (0 , ( z , ¯ z ) ¯ O (1 , ( z , ¯ z ) i vanish. So our AdS/CFT correspondenceis different from the usual one for scalar fields considered in [37][38]. This is thethird main result.We also examined HS [ ] × HS [ ] higher-spin gravity with fermionic matterfields, by extending the matter fields to odd polynomials of the auxiliary twistervariables y α . Two primary operators with spin are found, and their two-pointfunctions and three-point functions including spin-s current are obtained by ouron-shell action method.This paper is organized as follows. In sec.2 we will review our model for 3d higher-spin gravity coupled to 0-form ( C ) and two-form ( B ) matter fields. Its symmetry andgauge fixing procedure is explained. In sec.3 the equations of motion in the gauge-fixed form are solved. It is shown that the solution obtained here is also a solution tothe equations of motion in linearized 3d Vasiliev gravity. In sec.4 the action integralincluding the surface term is evaluated for this solution. This on-shell action has aform which can be eliminated by appropriate local boundary counterterm. In sec.5we choose the flat gauge fields A , ¯ A to be AdS , and evaluate the on-shell actionof sec.4 for various solutions in SL(3,R) × SL(3,R) higher-spin gravity. This on-shell action does not yield appropriate two- and three-point functions. In sec.6 wepropose a new boundary action, and show that this gives correct two- and three-point functions. In sec.7 our method is applied to HS [ ] × HS [ ] higher-spin gravity.For this purpose it is necessary to introduce new eigenfunctions of L = V withpositive eigenvalues. In addition to two known scalar operators, O (0 , , O (1 , , whichhave scaling dimensions ∆ = , , with their conjugates ¯ O (0 , , ¯ O (1 , , two spin operators, O (0 , , O (1 , with their conjugates are found. The two- and three-pointfunctions of these operators are computed by using the on-shell action. Summaryand discussion are given in sec.8 and three appendices follow. Some formulae forsl(3,R) are collected in appendix A, new formulae for Moyal products are givenin appendix B. In appendix C, scalar two-point functions in BTZ black hole arepresented. In this section we will review the model of 3d higher spin gravity coupled tomatter fields, which was introduced in [30]. This model is defined by an action5ntegral S CS-BC = S CS + S BC , where S CS is Chern-Simons (CS) action[16] S CS = k π Z M tr A ∧ (cid:16) dA + 23 A ∧ A (cid:17) − k π Z M tr ¯ A ∧ (cid:16) d ¯ A + 23 ¯ A ∧ ¯ A (cid:17) . (2.1)Here M = R × Σ is a 3d manifold and Σ is a 2d manifold with boundary ∂ Σ ∼ = S .In this section the properties of this model is explained by using spin-3 gravity asan example. Later in sec.7, the model extended to higher spin gravity based on hs [ ] ⊕ hs [ ] algebra will be considered. A = A µ dx µ and ¯ A = ¯ A µ dx µ are twoSL(3,R) gauge fields. They take values in sl(3,R) Lie algebra and can be expandedinto a basis of sl(3,R). These are related to vielbein e = e µ dx µ and spin connection ω = ω µ dx µ as A, ¯ A = ω ± ℓ e . ℓ is a constant related to the cosmological constant − ℓ . The level k in front of the two terms is related to the 3d Newton constant G as k = ℓ/ G . The second term of the action is given by S BC = Z M tr B ∧ (cid:16) dC + AC − C ¯ A (cid:17) . (2.2)This is the matter action and C is a real zero-form and B = B µν dx µ ∧ dx ν a realtwo-form field. Both fields are 3 × × SL(3,R) gaugetransformation, with U and ¯ U being SL(3,R) matrices, A → A ′ = U − dU + U − A U, ¯ A → ¯ A ′ = ¯ U − d ¯ U + ¯ U − ¯ A ¯ U ,C → C ′ = U − C ¯ U , B → B ′ = ¯ U − B U. (2.3)The diagonal SL(3,R) is local Lorentz transformation and the off-diagonal one islocal translation.[16][31][32]. In the pure spin-3 gravity case, the local translationcoincides with the ordinary diffeomorphism and the spin-3 transformation in themetric-like formalism. When the matter fields are coupled to spin-3 gravity, a sub-group of the local translation in the metric-like formalism does not coincide withdiffeomorphism.[30]In addition to these symmetries, there is a third one. Action S CS-BC has the Our notation for sl(3,R) algebra is collected in Appendix A. δ Ξ A = − πk (cid:18) C Ξ −
13 tr ( C Ξ) (cid:19) ,δ Ξ ¯ A = − πk (cid:18) Ξ C −
13 tr ( C Ξ) (cid:19) ,δ Ξ B = d Ξ + Ξ ∧ A + ¯ A ∧ Ξ ,δ Ξ C = 0 . (2.4)Here Ξ = Ξ µ dx µ is a one-form gauge parameter function which is also a 3 × δ Ξ A and δ Ξ ¯ A are introduced to ensurethe tracelessness of A and ¯ A . The transformation (2.4) with Ξ = d Σ + ¯ A Σ − Σ A ,where Σ is a zero-form, reduces on shell to SL(3,R) × SL(3,R) gauge transformation(2.3) with gauge parameters depending on C Σ and Σ C .[30]Let us review the Hamiltonian analysis in [30]. Since the action is first-order inderivatives, and is constructed as the integral of products of forms without an explicitmetric, it is already in a form of Hamiltonian. By singling out the time-componentsof fields, we rewrite S CS-BC as S CS-BC = Z M d x tr (cid:16) − k π ǫ ij A i ∂ t A j + k π ǫ ij ¯ A i ∂ t ¯ A j + 12 ǫ ij B ij ∂ t C (cid:17) + k π Z M d x tr (cid:16) A t ψ − ¯ A t ¯ ψ (cid:17) + Z M d x ǫ ij tr B ti χ j + k π Z ∂ M dt dφ tr (cid:16) A t A φ − ¯ A t ¯ A φ (cid:17) . (2.5)Here ǫ ij is a Levi-Civita symbol for the spatial components, and ǫ φρ = +1. Thecoordinates are t, φ and ρ . The last term is a boundary term on ∂ M = R t × S φ ,which appeared after partial integration. Functions ψ , ¯ ψ and χ i are defined as ψ = ǫ ij ( F ij + 8 πk C B ij − π k tr( CB ij )) , (2.6)¯ ψ = ǫ ij ( ¯ F ij + 8 πk B ij C − π k tr( CB ij )) , (2.7) χ i = ∂ i C + A i C − C ¯ A i . (2.8)Here F = dA + A ∧ A and ¯ F = d ¯ A + ¯ A ∧ ¯ A are field strengths.The momentum P conjugate to C is given by P ≡ ǫ ij B ij . The momentaconjugate to A i and ¯ A i are π iA ≡ k π ǫ ij A j and π i ¯ A ≡ − k π ǫ ij ¯ A j , respectively. Themomentum Π iB conjugate to B ti does not exist, and the primary constraint Π iB ≈ χ i ≈
0. Similarly, the momenta π A and π ¯ A A t and ¯ A t , respectively, obey π A ≈ π ¯ A ≈
0. These lead to secondaryconstraints, ψ ≈ ψ ≈
0. The Hamiltonian is a sum of Lagrange multiplierstimes these constraint functions.Constraints π A ≈ π ¯ A ≈ Π iB ≈ A t , ¯ A t , B ti , aswell as π A , π ¯ A , Π iB , are unphysical. So, by means of (2.4), we can gauge fix B tφ , B tρ such that B tφ = B tρ = 0. The two constraints ψ and ¯ ψ generate SL (3 , R ) × SL (3 , R )gauge transformations, and are first-class. The function χ i , the generator of the lo-cal transformation (2.4), transforms covariantly under these gauge transformations,hence χ i ≈ A i are eliminated by ψ ≈ A i .As for χ i , the role of these constraints is not to eliminate C , but to determine thederivatives of the field C in the spatial directions, φ and ρ . Corresponding to χ i , wethus propose to fix gauge by the conditions˜ χ i ≡ ∂ i P − P A i + ¯ A i P = 0 . (2.9)In this way, fields C and P are determined by their values at a sinlgle point inspace time. The number of constraints is larger than that of fields, and there are nophysical degrees of freedom. Combining the gauge fixing ˜ χ i ≈ P , we obtain the set of equations for P , ∂ µ P − P A µ + ¯ A µ P = 0 . (2.10)This provides the counterpart of the equations of motion for C , ∂ µ C + C A µ − ¯ A µ C =0. The equations of motion are given by F = − πk ( CB −
13 tr CB ) , (3.1)¯ F = − πk ( BC −
13 tr CB ) , (3.2) The analysis in this section also applies to HS[ λ ] × HS[ λ ] higher-spin gravity with 0 ≤ λ ≤
1. In this case, however, the internal space becomes infinite-dimensional. This means there arean infinite number of fields, and even in the presence of the constraint χ i = 0, there remains apropagating degree of freedom, tr C . In AdS space, tr C satisfies Klein-Gordon equation and has anon-polynomial solution with a delta-function boundary condition at space-like infinity. Recall that we set B tφ = B tρ = 0. The equation of motion for B µν is (3.4) below. d C + AC − C ¯ A = 0 , (3.3) d B + ¯ A ∧ B − B ∧ A = 0 . (3.4)As discussed in the previous section, we adopt the gauge fixing conditions B tρ = B tφ = 0 , ˜ χ i = ∂ i B φρ − B φρ A i + ¯ A i B φρ = 0 . (3.5)Then the equations of motion for the gauge fields are F tρ = 0 , (3.6) F tφ = 0 , (3.7) F φρ = − πk ( CB φρ −
13 tr CB φρ ) , (3.8)There are similar equations for ¯ A . The equations for B φρ are given by ∂ µ B φρ + ¯ A µ B φρ − B φρ A µ = 0 . (3.9)These are similar to those for C , ∂ µ C + A µ C − C ¯ A µ = 0 . (3.10)To solve these equations, we need to gauge fix A , ¯ A . This is done by the condi-tions A ρ = b − ( ρ ) ∂ ρ b ( ρ ) , (3.11)¯ A ρ = b ( ρ ) ∂ ρ b − ( ρ ) . (3.12)Here b ( ρ ) is a function of only ρ . This gauge fixing is always possible.[16]The equation for A t , ∂ ρ A t = − [ A ρ , A t ], which follows from (3.6), can now besolved to yield A t = b − ( ρ ) a t ( t, φ ) b ( ρ ) . (3.13) a t is an arbitrary function of t and φ . The equation for A φ , (3.8), can be rewrittenas ∂ ρ ( b A φ b − ) = 8 πk (cid:16) b C B φρ b − −
13 tr
C B φρ (cid:17) . (3.14)One can show that the righthand side is independent of ρ , because ∂ ρ ( bC b ) = 0due to (3.10) and ∂ ρ ( b − B φρ b − ) = 0 due to (3.9). Hence the solution to (3.14) isobtained as A φ = b − a φ ( t, φ ) b + 8 πk ( ρ − ρ ) (cid:16) C B φρ −
13 tr
C B φρ (cid:17) . (3.15)9ere ρ is a constant, which parametrize the solution, and a φ is an arbitrary functionof t and φ . The second term on the right corresponds to a torsion.[30] Finally, (3.7)follows, if a t and a φ satisfy an equation, ∂ t a φ − ∂ φ a t + [ a t , a φ ] (3.16)+ 8 π ( ρ − ρ ) k ∂ t ( b C B φρ b − −
13 tr
C B φρ ) + 8 π ( ρ − ρ ) k [ a t , b C B φρ b − ] = 0Now by using (3.10) we can show that ∂ t ( bCb ) + a t ( bCb ) − ( bCb ) ¯ a t = 0 . (3.17)Here ¯ a t is defined later in (3.30) by solving for ¯ A t as in (3.13). Combining this witha similar equation for B φρ , ∂ t ( b − B φρ b − ) + ¯ a t ( b − B φρ b − ) − ( b − B φρ b − ) a t = 0 , (3.18)which is obtained from (3.9), we find that ∂ t ( bCB φρ b − ) + a t ( bCB φρ b − ) − ( bCB φρ b − ) a t = 0 . (3.19)Trace of this equation leads to ∂ t tr ( CB φρ ) = 0. Hence the terms in the second lineof (3.16) cancel out: ∂ t a φ − ∂ φ a t + [ a t , a φ ] = 0 . (3.20)Therefore except for the extra term (3.15) in A φ , the solution for the gauge field isthe same as that in the pure spin-3 gravity obtained in [16]. In the following, wewill set a gauge condition a − ( t, φ ) ≡
12 ( a t ( t, φ ) − a φ ( t, φ )) = 0 . (3.21)Equation (3.20) is then satisfied, if a φ is holomorphic, a φ = a φ ( x + ): ∂ − a φ ≡
12 ( ∂ t − ∂ φ ) a φ = 0 . (3.22)Here we pause for a moment to consider the boundary conditions on the gaugefields. To make the variation problem well-defined, we must impose appropriateboundary conditions. When we vary the action with respect to A , ¯ A , B and C , weobtain boundary terms δ S CS-BC, boundary = − k π Z ∂ M tr ( A ∧ δ A − ¯ A ∧ δ ¯ A ) . (3.23) x ± ≡ t ± φ . R ∂ M tr B δ C drops, because B tφ = 0. In the pure spin-3 gravity,natural boundary conditions were [16] A − (cid:12)(cid:12) boundary = ¯ A + (cid:12)(cid:12) boundary = 0 . (3.24)Then the gauge fixing (3.21) ensures A − = 0 everywhere on shell.[16] When thematter fields are coupled, however, we obtain from (3.13) and (3.15) that A − = − πk ( ρ − ρ ) (cid:16) C B φρ −
13 tr
C B φρ (cid:17) . (3.25)This does not vanish as ρ → ∞ , and we cannot adopt the boundary conditions(3.24). Instead, we impose the following conditions. A − (cid:12)(cid:12) boundary → − πk ( ρ − ρ ) (cid:16) C B φρ − tr C B φρ (cid:17) , (3.26)( C, B φρ ) | boundary = fixed ( ρ → ∞ ) (3.27)As discussed at the end of sec.2, fields C and B φρ on the boundary are completelydetermined in terms of their values at a single point in the bulk by the covariant-constancy conditions (3.9) and (3.10). Hence the boundary condition (3.27) willbe appropriate. When the vielbein is computed from A and ¯ A , the spacetime isnot asymptotically AdS. Furthermore, we need to introduce surface terms on thetime-like boundary. S surface = k π Z ∂ M dt dφ tr (cid:0) A + A − + ¯ A + ¯ A − (cid:1) (3.28)Then the variation of S CS + S surface is given by k π R ∂ M dt dφ tr (cid:0) A + δ A − + ¯ A − δ ¯ A + (cid:1) .This vanishes, when the gauge field and matter fields are fixed as in (3.26) and (3.27).Parameter of gauge transformation which preserves the boundary condition (3.26)is given by Λ = b − λ ( x + ) b. (3.29)Since ∂ − λ = 0, A − transforms as δ A − = [ A − , Λ]. Because matters transform as δ C = − Λ C and δB φρ = B φρ Λ, the boundary condition (3.26) is covariant under(3.29). However, the condition (3.27) is not invariant.As for ¯ A , by repeating steps similar to those for A , we obtain (3.12) and¯ A t = b ¯ a t b − , (3.30)¯ A φ = b ¯ a φ ( t, φ ) b − + 8 πk ( ρ − ρ ) (cid:16) B φρ C −
13 tr
C B φρ (cid:17) . (3.31)11ere for simplicity we introduced the same integration constant ρ as in (3.15). ¯ a i must satisfy relations ¯ a φ = − ¯ a t , ∂ + ¯ a φ = 0 . (3.32)The boundary condition for ¯ A + is¯ A + (cid:12)(cid:12) boundary → πk ( ρ − ρ ) (cid:16) B φρ C −
13 tr
C B φρ (cid:17) . ( ρ → ∞ ) (3.33)Finally, we will solve for C and B φρ . By using (3.10) we have (3.17) and ∂ φ ( bCb ) + a φ ( bCb ) − ( bCb ) ¯ a φ = 0 , (3.34)where the terms of A φ and ¯ A φ which depend on C and B φρ canceled out. Eqs (3.17)and (3.34) yield the equations ∂ − ( b C b ) = − ( b C b ) ¯ a φ , (3.35) ∂ + ( b C b ) = − a φ ( b C b ) . (3.36)These equations are solved in terms of the ordered exponential ( P exp) and anti-ordered exponential ( P exp). C = b − P exp h − Z x + x +1 dx + a φ ( x + ) i! C (0) P exp h − Z x − x − dx − ¯ a φ ( x − ) i! b − (3.37)Here C (0) is a constant matrix, and x ± constants. In exactly the same manner,we obtain ∂ − ( b − B φρ b − ) = ¯ a φ b − B φρ b − and ∂ + ( b − B φρ b − ) = b − B φρ b − a φ ,which are solved for B φρ . B φρ = b P exp h Z x − x − dx − ¯ a φ ( x − ) i! B φρ (0) P exp h Z x + x +2 dx + a φ ( x + ) i! b (3.38)Hence solutions for C and B φρ are expressed in terms of flat gauge fields A µ ≡ b − a µ b + b − ∂ µ b and ¯ A µ ≡ b ¯ a µ b − + b ∂ µ b − . ( a ρ = ¯ a ρ = 0)To summarize, the solution obtained in this section also satisfies the followingequations. ( A ≡ A µ dx µ ) F ≡ d A + A ∧ A = 0 , (3.39)¯ F ≡ d ¯ A + ¯ A ∧ ¯ A = 0 , (3.40) d C + A C − C ¯ A = 0 , (3.41) d B φρ + ¯ A B φρ − B φρ A = 0 (3.42)12hese equations resemble those of 3d Vasiliev gravity with linearlized interaction.[40][12] There is a difference, however. In [12], field C is a complex scalar, anda conjugate of C is introduced and denoted as ¯ C . The holographic duals of thetraces of these fields, O and ¯ O , are conjugate to each other, and have non-vanishingtwo-point function. There also exists another complex scalar ˜ C , whose equationof motion is same as (3.42) for B φρ . There also exists a conjugate of ˜ C , i.e. , h C ,and the traces of both fields have holographic duals, ˜ O and h O , with non-vanishingtwo-point function. In our model, C ours and B φρ are real fields, and B φρ plays therole of both ˜ C and ¯ C . It might seem that, to make the correspondence of the twomodels exact, we need to double our real scalars. It will, however, be shown in sec.7that it is possible to obtain all necessary two-point functions in our model withoutintroducing extra matter fields. In AdS/CFT correspondence, a generating functional of the correlation functionson the boundary CFT is obtained by substituting solutions to the equations ofmotion into the action integral S CS-BC + S surface .[35][36][37] Let us compute thison-shell action.When the solutions to the equations of motion are substituted, S BC (2.2) van-ishes. As for S CS (2.1), by rewriting dA + A ∧ A as F − A ∧ A and using (3.1),we obtain S CS (on shell) = k π Z tr A ∧ (cid:0) − πk CB − A ∧ A (cid:1) − k π Z tr ¯ A ∧ (cid:0) − πk BC −
13 ¯ A ∧ ¯ A (cid:1) . (4.1)By using (3.5) this is rewritten as − R M tr ( A t C B φρ ) d x − k π R M tr [ A t , A φ ] A ρ d x +(similar terms for ¯ A ). Owing to (3.7), the integrand of the second term becomestotal derivative terms, tr ( ∂ φ A t − ∂ t A φ ) A ρ ( ρ ) = tr (cid:0) ∂ φ ( A t A ρ ) − ∂ t ( A φ A ρ ) (cid:1) , hencethe integral vanishes. We obtain S CS (on shell) = − ρ c Z ρ = ρ c dt dφ tr ( A t C B φρ ) + 12 ρ c Z ρ = ρ c dt dφ tr ( ¯ A t B φρ C ) . (4.2)Here we used the fact that the trace terms do not depend on ρ , and the ρ integra-tions amount to simply multiplying the integrands by ρ c , where ρ c is a large cut-offrepresenting the location of the boundary. 13he surface term (3.28) on shell is given by S surface (on shell) = −
12 ( ρ c − ρ ) Z ρ = ρ c dt dφ tr ( a φ b C B φρ b − ) −
12 ( ρ c − ρ ) Z ρ = ρ c dt dφ tr (¯ a φ b − B φρ C b ) − πk ( ρ c − ρ ) Z ρ = ρ c dt dφ (cid:0) tr( B φρ C ) −
13 (tr B φρ C ) (cid:1) . (4.3)By adding (4.2) and (4.3), we obtain the on-shell action Z ρ = ρ c dtdφ (cid:16) − ( 12 ρ − ρ c ) tr (cid:0) B φρ ∂ + C ) − ( 12 ¯ ρ − ρ c )tr (cid:0) B φρ ∂ − C (cid:1)(cid:17) − πk ( ρ c − ρ ) Z ρ = ρ c dt dφ (cid:0) tr( B φρ C ) −
13 (tr B φρ C ) (cid:1) . (4.4)Here (3.37) and (3.38) are used. This is divergent as ρ c → ∞ , but the divergencecan be cancelled by local boundary counterterms S counterterms = − ( ρ c − ρ ) Z ρ = ρ c dtdφ tr (cid:0) B φρ ∂ + C ) − ( ρ c − ¯ ρ ) Z ρ = ρ c dt dφ tr (cid:0) B φρ ∂ − C (cid:1) + 4 πk ( ρ c − ρ ) Z ρ = ρ c dt dφ (cid:0) tr( B φρ C ) −
13 (tr B φρ C ) (cid:1) . (4.5) ρ and ¯ ρ are some finite constants. Finally, the on-shell action which includes thecounterterms becomes finite. It can be re-expressed in a simpler form. S on shell = µ Z ρ = ρ c dtdφ tr (cid:0) B φρ ∂ + C ) + ¯ µ Z ρ = ρ c dt dφ tr (cid:0) B φρ ∂ − C (cid:1) , (4.6)where µ = ρ − ρ and ¯ µ = ¯ ρ − ρ . The quartic terms are eliminated, becausethe theory is free. By using (3.22), (3.35) and a similar equation for b − B φρ b − ,it can be shown that the integrand of the first term of (4.6) is independent of x − .Similarly, that of the second term does not depend on x + , either.There remains ambiguity in the finite coefficients in front of the above two in-tegrals. Notice that by setting ρ = 2 ρ and ρ = 2 ¯ ρ , we can drop both terms.In sec.5 we will show that this boundary action does not yield appropriate corre-lation functions which satisfy W-algebra Ward identities. We will then drop thison-shell action completely by using the local counterterms, and introduce anotherlocal boundary action integral in sec.6. In this way the residual gauge symmetry(3.29) will be recovered. 14 Explicit Solution for C and B in AdS Backgrounds
As we have seen, although the gauge fields A , ¯ A are not flat in the matter-coupled theory, the fields a , ¯ a ( A µ , ¯ A µ ) are flat, and it may make sense to discussmatter fields associated to AdS background. As a warming up, in this section wewill compute the solutions for these fields in AdS , and evaluate the on-shell action(4.6) in SL(3,R) × SL(3,R) theory. In sec.7 higher spin gravity with HS [ ] × HS [ ]gauge symmetry will be considered.For asymptotically AdS spacetime, a and ¯ a are given by [43] a = ( L + 2 πk L ( x + ) L − ) dx + , (5.1)¯ a = − ( L − + 2 πk ¯ L ( x − ) L ) dx − , (5.2)and b ( ρ ) by b ( ρ ) = e ρ L . (5.3)For matrices L i see appendix A. The AdS spacetime in the Poincar´e patch is givenby L = ¯ L = 0, and the one in the global patch is obtained by the choice πk L = πk ¯ L = . In what follows we will consider AdS spacetime in the Poincar´e patch.So we set a = L dx + and ¯ a = − L − dx − , and the metric is given by ds = ℓ ( dρ + e ρ ( − dt + dφ )).We Wick rotate the spacetime to Euclidean AdS , by replacements x + = t + φ → φ − iτ = z , x − = t − φ → − ( φ + iτ ) = − ¯ z . From (3.37) and (3.38) we write downthe formulae for C and B φρ . C ( z, ¯ z, ρ ) = e − ρ L e − ( z − z ) L C (0) e (¯ z − ¯ z ) L − e − ρ L , (5.4) B φρ ( z, ¯ z, ρ ) = e ρ L e − (¯ z − ¯ z ) L − B φρ (0) e ( z − z ) L e ρ L . (5.5)Here z , , ¯ z , are complex numbers which specify the locations of CFT operators.We checked that the components of C B φρ and B φρ C behave at most as e ρ for ρ → + ∞ . For the two operators dual to the sources C and B φρ , respectively, tohave same conformal weights, constant matrices C (0) and B φρ (0) must satisfy thefollowing pairing rule for eigenvalues of L . L C (0) = − h C (0) , C (0) L = − h ′ C (0) ,L B φρ (0) = h ′ B φρ (0) , B φρ (0) L = h B φρ (0) (5.6)15n this case, ρ -dependence of both C ( z, ¯ z, ρ ) and B φρ ( z, ¯ z, ρ ) is C ( z, ¯ z, ρ ) = e ( h + h ′ ) ρ C ( e ρ z, e ρ ¯ z, B φρ . There are 9 pairs to take into account. Out of themthe following three yield non-vanishing on-shell actions. (1.) C (1) (0) = , B (1) φρ (0) = , ( h, h ′ ) = ( − , − (2.) C (2) (0) = , B (2) φρ (0) = , ( h, h ′ ) = ( − , (3.) C (3) (0) = , B (3) φρ (0) = , ( h, h ′ ) = (0 , − C ( z, ¯ z, ρ ) and B φρ ( z, ¯ z, ρ ), and then (4.6) gives theon-shell action. The above choices for C (0) and B (0) lead to the following on-shellaction integrals. (1.) S (1)on shell = Z d z (cid:16) − µ ( z − z )(¯ z − ¯ z ) + 2¯ µ ( z − z ) (¯ z − ¯ z ) (cid:17) (5.7) (2.) S (2)on shell = − µ Z d z ( z − z ) (5.8) (3.) S (3)on shell = 4 ¯ µ Z d z (¯ z − ¯ z ) (5.9)Note that the integrands do not depend on z and ¯ z . As mentioned at the end of sec.4,the two integrands of (4.6) do not depend on ¯ z and z , respectively. Furthermore,the above integrands do not depend on both coordinates, because a + and ¯ a − areconstant for AdS . These on-shell actions would be expected to give two-pointfunctions. If the coefficients of both terms in S (1)on shell were non-zero, then the two-point function does not have a suitable form, ( z − z ) − h (¯ z − ¯ z ) − h . We must16et either coefficient to zero by adjusting the parameters of the solutions; ρ =2 ρ or ¯ ρ = 2 ¯ ρ . All in all, there would be four operators of conformal weights( h, ¯ h ) = ( − , − ) , ( − , − , (0 , − ) and ( − , h, ¯ h ) =( − , − C [13], is missing. Inthe on-shell action S (1)on shell , the two terms have forms ∂ z (cid:0) ( z − z ) (¯ z − ¯ z ) (cid:1) and ∂ ¯ z (cid:0) ( z − z ) (¯ z − ¯ z ) (cid:1) , respectively. The derivative ∂ z in the first term comesfrom an a + = L insertion in (4.6),tr (cid:0) a + b C B φρ b − (cid:1) = tr (cid:16) L e ( z − z ) L C (0) e (¯ z − ¯ z ) L − B φρ (0) (cid:17) = ∂ z (cid:16) tr e ( z − z ) L C (0) e (¯ z − ¯ z ) L − B φρ (0) (cid:17) . (5.10)This is again a result of the fact that a + is a constant matrix. Similarly, in the secondterm, ∂ ¯ z comes from ¯ a − = − L − insertion. If there were no a + , ¯ a − insertions, thenthe spectrum of the conformal weights would be ( h ′ , ¯ h ′ ) = ( − , − − ,
0) and(0 , − C ( z, ¯ z, ρ ) and B φρ ( z, ¯ z, ρ ) in the three cases (1.), (2.), (3.). For the firstsolution we have tr C (1) = e − ρ { e ρ ( z − z )(¯ z − ¯ z ) } , (5.11)tr B (1) φρ = e − ρ { e ρ ( z − z )(¯ z − ¯ z ) } . (5.12)These may be formally taken as regularization of delta function e − ρ δ (2) ( z − z i )[13],although the powers on the right hand sides have wrong signs to represent deltafunctions. The others are derivatives of the first one: tr C (2) = − ∂ ¯ z tr C (1) ,tr C (3) = − ∂ z tr C (1) . tr B (2) φρ = ∂ ¯ z tr B (1) φρ , tr B (3) φρ = ∂ z tr B (1) φρ . Because all compo-nents of C are related to tr C [12], all the four solutions are connected. Hence it isexpected that the operators O ( i ) , ¯ O ( i ) ( i = 2 ,
3) are descendants of O (1) and ¯ O (1) .The two-point functions of these operators can be computed from S ( i )new bdry (i=2,3)separately.We now turn to computation of three-point correlation functions involving twooperators dual to the solutions C (1) and B (1) , and one higher-spin current, < O ( z ) ¯ O ( z ) J ( s ) ( z ) > . For brevity, from now on, the operator dual to C (1) willbe denoted as O , and the one dual to B (1) φρ as ¯ O .To compute the correlation functions, we need to deform the spacetime fromAdS slightly. The deformation of the metric is dual to the stress tensor, and the17igher-spin gauge field the higher-spin current. Since the gauge fields a i and ¯ a i areflat, these deformations can be achieved by gauge transformation. C ( z, ¯ z, ρ ) → b − U ( z ) b C ( z, ¯ z, ρ ) , (5.13) B φρ ( z, ¯ z, ρ ) → B φρ ( z, ¯ z, ρ ) b − U − ( z ) b (5.14)For an infinitesimal transformation, we write U ( z ) = 1 + Λ( z ). However, the abovetransformations are slightly incorrect. Since the solutions (3.37) and (3.38) areordered exponentials, the correct transformations are, by using (5.4) and (5.5), C ( z, ¯ z, ρ ) → e − ρ L U ( z ) e − ( z − z ) L U − ( z ) C (0) e (¯ z − ¯ z ) L − e − ρ L , (5.15) B φρ ( z, ¯ z, ρ ) → e ρ L e − (¯ z − ¯ z ) L − B φρ (0) U ( z ) e ( z − z ) L U − ( z ) e ρ L . (5.16)Let us consider the first term of the on-shell action (4.6), S on shell 1 = Z d z tr B φρ ∂ z C. (5.17)The variation of the above action to order O (Λ ) will give the three-point function. < O ( z ) ¯ O ( z ) J ( s ) ( z ) > ≡ δ S on shell 1 (5.18)The variation is given by δ S on shell 1 = Z d z tr C (0) e (¯ z − ¯ z ) L − B φρ (0) e ( z − z ) L ∂ z Λ( z ) e ( z − z ) L + Z d z tr n(cid:0) e ( z − z ) L L Λ( z ) − Λ( z ) L e ( z − z ) L (cid:1) C (0) e (¯ z − ¯ z ) L − B φρ (0) o . (5.19)The integrand of the first term depends on z , while that of the second term doesnot. The form of Λ( z ) is given by [12] Λ( z ) = s − X n =1 n − − ∂ z ) n − Λ ( s ) ( z ) V ss − n . (5.20)Here V sm is an spin-s generator of higher-spin algebra, and especially, V m = L m and V m = W m in the case of sl (3 , R ). In CFT the three-point function < O ( z ) ¯ O ( z ) J ( s ) ( z ) > of primary operators O and ¯ O must have a form < O ( z ) ¯ O ( z ) J ( s ) ( z ) > ∝ (cid:16) z − z ( z − z )( z − z ) (cid:17) s < O ( z ) ˜ O ( z ) > . (5.21) The following presctiption is based on the idea in [12]. Here the gauge transformation (5.13)-(5.14) is applied to the boundary term, not to the propagator. This is related to Λ( ρ, z ) in (4.16) of [12] by Λ( z ) = e ρ V Λ( ρ, z ) e − ρ V . For W m , see appendix A.
18e consider the case of spin 2 (s=2) transformation, and set Λ (2) ( z ) = π ( z − z ) .When C (1) and B (1) φρ for the solution (1.) is substituted into the second term of(5.19), we have Z d z ( z − z )(¯ z − ¯ z ) π ( z − z ) ( z − z ) { ( z − z ) + ( z − z ) − ( z − z )( z − z ) } . (5.22)Since < O ( z ) ¯ O ( z ) > = ( z − z )(¯ z − ¯ z ) , this does not agree with (5.21) with s = 2. There also exists the extra first term in (5.19). Hence the correlation functionsin (5.7) are not those of primary operators. Similar analysis to the cases (2.)-(3.)shows that the correlation functions (5.8)-(5.9) are not, either. With the observation in the previous section we are forced to set the on-shellaction (4.6) to zero by adjusting the parameters of the local counterterms in sucha way that µ = ¯ µ = 0. Then the residual gauge symmetry (3.29) is recovered.In order to obtain the generating functional of the correlation functions of CFToperators, we need to introduce new appropriate boundary terms. In this contextthe example[33][34] of a free spinor is helpful. In AdS/CFT correspondence for afree spinor ψ , a boundary term R ∂ M d x √ γ ¯ ψψ is added to the bulk action, becausethe bulk action vanishes when a solution to the equation of motion is substituted.This fermion boundary term keeps all the symmetry required.A simplest prescription would be to adopt the surface term, which appears inthe action of a free scalar field in AdS background, after substitution of the solutionto the equation of motion and partial integration. Since tr C obeys Klein-Gordonequation with mass m = λ − λ = 3[12], a term like R ∂ M d x √ γ tr B φρ tr C ,where γ µν = e aµ e aν | ∂M is the induced metric on the boundary, is expected to work.Here, a derivative on tr C with respect to ρ is not introduced. Such a derivativewill simply modify the generating functional by a multiplicative constant. It can beshown that this surface term give an appropriate generating functional for two-pointfunctions. In the case of HS[ ] × HS[ ] higher-spin gravity which will be treatedin sec.7, however, this prescription for calculating two-point functions would givethose for operators of scaling dimension ∆ + = or ∆ − = only. Moreover, thisboundary term breaks the residual gauge symmetry (3.29). Hence, in the remainderof this paper, we will not exploit this boundary term. γ ij is an induced metric on ∂ M .
19e propose the following new local boundary term. S new bdry = lim ρ →∞ Z d z g A tr (cid:0) B φρ ( z, ¯ z, ρ ) C ( z, ¯ z, ρ ) (cid:1) (6.1)Here g is a constant different from zero. A is the area of the boundary. Thisboundary term will be added to the action integral from the beginning. Additionof a new boundary term modifies the theory. The relative coefficient between theterm (6.1) and the other part of the action (2.1)+(2.2)+ (3.28) is not determinedin the present context, and we will simply set the coefficient of (6.1) to a non-vanishing constant g . The above term (6.1) is invariant under the residual gaugetransformation (3.29). When the solution to the equations of motion is substituted,this trace is independent of ρ . Hence actually, the limit ρ → ∞ is not necessary.The reason for this peculiar phenomenon is that the equation of motion for C makesthe solution a parallel transport of its value at a single point, and that (6.1) is gaugeinvariant. The data of the fields are transfered to the internal space. Hence the term(6.1) captures the property of the matter fields more properly. It will be shown thatthe above boundary term (6.1) works as a correct generating functional for two-pointfunction and three-point functions. The integrand is also independent of z and ¯ z ,so that the area A of the boundary will be cancelled out. So (6.1) can be replacedby S new bdry = g tr (cid:0) B φρ ( z, ¯ z, ρ ) C ( z, ¯ z, ρ ) (cid:1) .When the solution (1.) in sec.5 is substituted into (6.1), we obtain the followingon-shell action. S (1)new bdry = g ( z − z ) (¯ z − ¯ z ) (6.2)The two-point function of the operators O and ¯ O dual to C (1) and B (1) are given by < O ( z ) ¯ O ( z ) > = ( z − z ) (¯ z − ¯ z ) and the conformal weights ( h, ¯ h ) are ( − , − C and B φρ with extra variations at z and z , the ‘invariant’ boundary action (6.1) transforms as δ S new bdry = g tr n e (¯ z − ¯ z ) L − B φρ (0) (cid:18) Λ( z ) e ( z − z ) L − e ( z − z ) L Λ( z ) (cid:19) C (0) o . (6.3)For spin-2 transformation with Λ( z ) = π ( z − z L + z − z ) L + z − z ) L − ), thevariation of the action with the solution (1.) substituted, is given by δ S (1)new bdry = − g π (cid:16) z − z ( z − z )( z − z ) (cid:17) ( z − z ) (¯ z − ¯ z ) . (6.4) In the case of a Klein-Gordon scalar field φ ( x ) in AdS background, the integrand of the boundaryaction which appears after partial integration, φ ( x ) ∂ ρ φ ( x ), depends on ρ . z ) = π ( z − z W + z − z ) W + z − z ) W + z − z ) W − + z − z ) W − ), we obtain δ S (1)new bdry = 13 π (cid:16) z − z ( z − z )( z − z ) (cid:17) S (1)new bdry . (6.5)Hence the operators O and ¯ O dual to C (1) and B (1) φρ are primary ones. H S [ ] × H S [ ] CS theory
The above analysis can be extended to the 3d higher-spin gravity based on HS [ λ ] × HS [ λ ] gauge symmetry.[11] In the action integral S CS − BC , product of matri-ces A µ , ¯ A µ , C and B µν must be simply replaced by their lone-star product ⋆ [41][12].When the parameter λ is equal to , this product reduces to a Moyal product ∗ ,and the calculation simplifies.[12] In what follows, we will restrict discussion only tothis case. The Moyal product of two functions is defined by( f ∗ g )( y ) = 14 π Z d u d v f ( y + u ) g ( y + v ) e iuv . (7.1)Here y α , u α and v α ( α = 1 ,
2) are twister variables, with uv = ǫ αβ u α v β , and ǫ = 1.The ∗ -commutator of y α is given by [ y , y ] ∗ = 2 i . The generators of hs [ ] are definedby even polynomials of y α . V sm = (cid:18) − i (cid:19) s − y s + m − y s − m − . ( s = 2 , , . . . ; 1 − s ≤ m ≤ s −
1) (7.2)Especially, s=2 generators satisfy the algebra of L i ’s: [ V , V ] ∗ = V , [ V , V − ] ∗ =2 V and [ V , V − ] ∗ = V − . The trace operation is replaced bytr y f ( y , y ) = f (0 , . (7.3)In this section the matter fields are extended to include odd polynomials of y α .They are split as C = C e + C o , B φρ = B e φρ + B o φρ , (7.4)where the fields with a superscript e or o are made of even and odd polynomials,respectively. C e and B e φρ are bosons. C o and B o φρ represent fermionic fields, sincetr y B o φρ C o = − tr y C o B o φρ , due to a formulatr y f ( y ) ∗ g ( y ) = tr y g ( − y ) ∗ f ( y ) = tr y g ( y ) ∗ f ( − y ) . (7.5)When (7.4) is substituted into the action (2.2), the action for the bosonic fields andthe fermionic ones decouple, because Moyal product keeps parity of polynomials andthe trace of odd polynomials vanish. 21 .1 New Eigenfunctions of V The eigenfunctions of V are given by[11] f mn ( y ) ≡ y m ∗ e − iy y ∗ y n . ( m, n = 0 , , . . . ) (7.6)These functions satisfy V ∗ f mn ( y ) = − m + 14 f mn ( y ) ,f mn ( y ) ∗ V = − n + 14 f mn ( y ) . (7.7)This fact (for m = n = 0 ,
1) was used in [11] to construct two scalar functions Tr C ± .However, in this case the eigenvalues of (7.7) are all negative. This does not fit toour purpose.Let us define another set of eigenfunctions ˜ f mn ( y ) by˜ f mn ( y ) ≡ y m ∗ e iy y ∗ y n . ( m, n = 0 , , . . . ) (7.8)It can be shown that this satisfies the equations. V ∗ ˜ f mn ( y ) = 2 m + 14 ˜ f mn ( y ) , ˜ f mn ( y ) ∗ V = 2 n + 14 ˜ f mn ( y ) . (7.9)(7.6) and (7.8) allow us to define pairs of C (0) and B φρ (0) that obey (5.6). Thereare two types of assignments of eigenfunctions f mn and ˜ f mn . A. C ( m,n ) (0) = f mn ( y ) , B ( n,m ) φρ (0) = ˜ f nm ( y ) . ( m, n = 0 , , . . . ) (7.10) B. C ( m,n ) (0) = ˜ f mn ( y ) , B ( n,m ) φρ (0) = f nm ( y ) . ( m, n = 0 , , . . . ) (7.11) The new boundary action (6.1) for this model is given by the following twotraces. S ( A,mn )new bdry = g tr y (cid:16) y n ∗ e iy y ∗ y m ∗ e − i ( z − z ) y ∗ y m ∗ e − iy y ∗ y n ∗ e i (¯ z − ¯ z ) y (cid:17) , (7.12) S ( B,mn )new bdry = g tr y (cid:16) y n ∗ e − iy y ∗ y m ∗ e − i ( z − z ) y ∗ y m ∗ e iy y ∗ y n ∗ e i (¯ z − ¯ z ) y (cid:17) (7.13)22hese on-shell actions can be computed by using the method in [11]. Some newformulae are presented in appendix B. By differentiating (B.3) with respect to z and ¯ z necessary times, we obtain S ( A,m,n )new bdry . S ( A,mn )new bdry = 12 g ( − i ) n + m (2 m )! (2 n )! m ! n ! ( z − z ) − − m (¯ z − ¯ z ) − − n (7.14)Here formula (7.5) is used to move y n in the middle of the trace to the leftmost.Note that the limit ρ → ∞ is not taken.The second surface on-shell action S ( B,mn )new bdry is ill-defined: by using (B.2) oneobtainstr y e − i zy ∗ e ξ y ∗ e iy y ∗ e η y ∗ e i ¯ zy ∗ e ξ y ∗ e − iy y ∗ e η y = e i ( ξ η − ξ η )+ iz ( ξ ) − i ¯ z ( ξ ) tr y e iy y +2( η y + ξ y ) ∗ e − iy y +2( ξ y + η y ) (7.15)Here η i and ξ i are sources for y i , and z = z − z , ¯ z = ¯ z − ¯ z . When (B.2) is againapplied to the Moyal product in the last line, one obtains a divergent result. Thisdivergence cannot be removed by an overall renormalization of the solutions. Evenif this divergence is regularized temporally, it does not lead to conformally covarianttwo-point functions. This asymmetry in C and B occurs, because in (5.2), a + isassociated with L and ¯ a − with L − . Similarly, in the spin-3 case in secs.4 and 5,the on-shell action completely vanishes for C being a right-lower triangular matrixand B a left-upper triangular one. We will not consider this type of solutions. Let us now identify primary operators. The A-type solutions C ( m,n ) ( z, ¯ z, ρ ) = e − ρ V ∗ ∗ e − ( z − z ) V ∗ f mn ( y ) ∗ e (¯ z − ¯ z ) V − ∗ e − ρ V ∗ can be classified into four sets ac-cording to the values of m and n (mod 2 Z ). C ( ν +2 a,ν +2 b ) ( z, ¯ z, ρ )= e − ρ V ∗ ∗ e i ( z − z ) y ∗ y ν +2 a ∗ e − iy y ∗ y ν +2 b ∗ e − i (¯ z − ¯ z ) y ∗ e − ρ V ∗ = e ( h +¯ h ) ρ (cid:16) e i ( z − z ) e ρ y ∗ y ν +2 a ∗ e − iy y ∗ y ν +2 b ∗ e − i e ρ (¯ z − ¯ z ) y (cid:17) . (7.16)Here ν , ν = 0 , a, b = 0 , , , · · · and( h, ¯ h ) = (cid:16) ν + 14 + a, ν + 14 + b (cid:17) (7.17)23s the conformal weight. Since y a and y b can be replaced by ( ∂ z ) a and ( ∂ ¯ z ) b , respec-tively, up to multiplying constants, the solutions with a ≥ b ≥ C ( ν ,ν ) which are dual to the primaryoperators.Solutions C (0 , and C (1 , are already obtained in [11] and denoted as C − and2 iC + . C (0 , ( z, ¯ z, ρ ; z , ¯ z ) = e ρ p | L | e y T S y , (7.18) C (1 , ( z, ¯ z, ρ ; z , ¯ z ) = e ρ p | L | e y T S y n i | L | + 4 | L | ( y + (¯ z − ¯ z ) e ρ y ) ( y − ( z − z ) e ρ y ) o (7.19)Here | L | and S have the following expressions. | L | = e ρ | z − z | + 1 , (7.20) S = i | L | (cid:18) e ρ ( z − z ) | L | − | L | − − e ρ (¯ z − ¯ z ) (cid:19) . (7.21)The traces of these fields formally behave for ρ → ∞ astr y C (0 , ∼ − πe − ρ δ (2) ( z − z ) + e − ρ | z − z | − , (7.22)12 i tr y C (1 , ∼ π e − ρ δ (2) ( z − z ) + e − ρ | z − z | − . (7.23)In analogy with the usual AdS/CFT correspondence for a scalar field φ ( x ) [37][38],these traces are expected to be sources of scalar operators in CFT on the boundary.There is, however, a difference between the BC model coupled to higher spin gravityand the ordinary scalar field theory. It is known that there are two ways to quantizea scalar field in AdS [39], and there are corresponding operators O + and O − . In theusual AdS/CFT correspondence, the scalar field φ works as a source for O + , andthe generating functional of correlation functions of O − is obtained by Legendretransformation of that of O + . In the present case, operators dual to tr C (0 , and i tr C (1 , are denoted as O (0 , and O (1 , , respectively. Similarly, operators dualto tr B (0 , φρ and i tr B (1 , φρ are denoted as ¯ O (0 , and ¯ O (1 , . Then the two-pointcorrelation functions of these operators are obtained from (7.14) straightforwardly(without ρ → ∞ limit and Legendre transformation) as < O (0 , ( z ) ¯ O (0 , ( z ) > = 12 g ( z − z ) − / (¯ z − ¯ z ) − / , (7.24) < O (1 , ( z ) ¯ O (1 , ( z ) > = 12 g ( z − z ) − / (¯ z − ¯ z ) − / . (7.25)24his point is in sharp contrast to the usual AdS/CFT correspondence in a free scalartheory[38]. All primary operators can be quantized in a single quantization. Solutions C (0 , and C (1 , do not belong to hs [ ], because C (0 , ( z, ¯ z, ρ ; z , ¯ z ) = e ρ | L | e y T S y (cid:0) y − e ρ ( z − z ) y (cid:1) , (7.26) C (1 , ( z, ¯ z, ρ ; z , ¯ z ) = e ρ | L | e y T S y (cid:0) y + e ρ (¯ z − ¯ z ) y (cid:1) . (7.27)They are made up of terms with odd number of y ’s, and are fermions. These havespinor components.tr y C (0 , ∗ y = − i | L | e ρ ( z − z ) , tr y C (0 , ∗ y = − i | L | e ρ , (7.28)tr y C (1 , ∗ y = − i | L | e ρ (¯ z − ¯ z ) , tr y C (1 , ∗ y = 2 i | L | e ρ . (7.29)The components of C (01) behave for ρ → ∞ astr y C (01) ∗ y ∼ − πi e − ρ ∂ ¯ z δ (2) ( z − z ) − i e − ρ ( z − z ) − (¯ z − ¯ z ) − , tr y C (01) ∗ y ∼ − πi e − ρ δ (2) ( z − z ) − i e − ρ ( z − z ) − (¯ z − ¯ z ) − . (7.30)Solution to the equation of motion for a free massive Dirac fermion in AdS with achiral boundary condition is given by[33][34] ψ ( x , ρ ) = Z d x (cid:16) e − ρ Γ +( x − x ) · Γ (cid:17) (cid:16) e − ρ + | x − x | (cid:17) − + m Γ e ρ ( − m Γ ) ψ ( x ) . (7.31)Here Γ = − i Γ Γ is a gamma matrix in the radial direction, and Γ = (Γ , Γ ), thosein the φ, t directions. For a chiral boundary condition which imposes Γ ψ = + ψ ,one has ( x − x ) · Γ ψ = ( z − z ) Γ ψ . By comparing this expression (7.31) withthe two components (7.28), one can identify, if one sets m = 0, ψ ( z, ¯ z, ρ ) = i Z d z tr y ( C (01) ( z, ¯ z, ρ ) ∗ ( y − y Γ )) ψ ( z , ¯ z ) . (7.32)The components of C (10) (7.29) can also be related to anti-chiral spinor ψ withΓ ψ = − ψ . The operators dual to tr ( C (01) ∗ ( y − y Γ )) and tr ( C (10) ∗ ( y + y Γ ))will be denoted as O (0 , and O (1 , , respectively. The solutions B ( ν ,ν ) φρ can also becomputed, and ¯ O ( ν ,ν ) are defined similarly. Two-point correlation functions ofthese operators are obtained from (7.14) as < O (0 , ( z ) ¯ O (1 , ( z ) > = − i g ( z − z ) − / (¯ z − ¯ z ) − / ,< O (1 , ( z ) ¯ O (0 , ( z ) > = − i g ( z − z ) − / (¯ z − ¯ z ) − / . (7.33) For off-diagonal two-point functions, see the end of this subsection. h, ¯ h ) = ( , ) and ( , ), respectively. These havespin h − ¯ h = ± . The scaling dimension is given by ∆ ≡ h + ¯ h = 1. Comparingthis with the result of [33], ∆ = d + m with d = 2, we again find that the fermionis massless, m = 0.One can show that off-diagonal two-point functions vanish: let us take linearcombinations of the independent solutions C (0 , and C (1 , in (7.22) and (7.23). C ( z, ¯ z, ρ ) = − π Z d z C (0 , ( z, ¯ z, ρ ; z , ¯ z ) φ ( z , ¯ z ) − πi Z d z C (1 , ( z, ¯ z, ρ ; z , ¯ z ) φ ( z , ¯ z ) (7.34)Here φ ( z , ¯ z ) and φ ( z , ¯ z ) are boundary conditions for the two modes. Similarlyfor B φρ : B φρ ( z, ¯ z, ρ ) = − π Z d z B (0 , φρ ( z, ¯ z, ρ ; z , ¯ z ) ¯ φ ( z , ¯ z ) − πi Z d z B (1 , φρ ( z, ¯ z, ρ ; z , ¯ z ) ¯ φ ( z , ¯ z ) (7.35)Then the boundary term tr B φρ C is given by18 π Z d z Z d z φ ( z , ¯ z ) ¯ φ ( z , ¯ z ) 1 | z − z | + 18 π Z d z Z d z φ ( z , ¯ z ) ¯ φ ( z , ¯ z ) 1 | z − z | (7.36)We also obtain a similar result for fermionic solutions, and we conclude that < O ( ν ,ν ) ( z , ¯ z ) ¯ O ( ν ′ ,ν ′ ) ( z , ¯ z ) > = 0 , if ( ν , ν ) = ( ν ′ , ν ′ ) . (7.37)Hence, in our prescription we can consider all the primary operators, O ( ν ,ν ) and¯ O ( ν ,ν ) , in a single quantization. In the usual AdS/CFT correspondence of the scalarfield[37][38], only operator O (1 , can be considered in the standard quantization, andonly O (0 , in the alternate quantization. In the minimal model holography[5], twooperators O (0 , and O (1 , are expected to work as ladder operators to generateother primary operators by fusion, and two complex scalar fields are introduced in[5]. In our formalism it is possible to introduce both primary operators by meansof a single field C . It would be interesting if there were a formalism, where in atheory of a single real bulk scalar field φ of mass − d < m < − d in AdS d +1 , B (0 , φρ ( z, ¯ z, ρ ; z , ¯ z ) and B (0 , φρ ( z, ¯ z, ρ ; z , ¯ z ) are the counterparts of (7.22) and (7.23). If two sets of fields C and B were introduced, then the symmetry transformation (2.4) couldnot be extended to the two sets. ± = d ± q d + m appear in a boundaryCFT. Actually, there are several quantizations for scalar fields in AdS.[46] Difficulty,however, lies in the difference of the relative signs of the two terms in (7.22) and(7.23). This might lead to non-unitarity of CFT. In the calculation of the two-point functions presented in this paper this problem does not occur, because theasymptotic behaviors (7.22) and (7.23) are not used.In appendix C, two scalar propagators on BTZ black hole[47] at λ = are studiedand it is shown that the results in the literature[11] are obtained by means of ourboundary action (6.1). Furthermore, two-point functions of operators with differentscaling dimensions vanish. Hence our prescription also works in backgrounds morecomplicated than AdS vacuum. We will compute the three-point functions in order to check the operators dis-cussed in the previous subsection are not just quasi-primary, but really primary ones.These correlation functions are obtained by gauge transformation of the boundaryaction S A,ν ,ν new bdry . < O ( ν ,ν ) ( z ) ¯ O ( ν ,ν ) ( z ) J ( s ) ( z ) > (7.38)= g tr y n e (¯ z − ¯ z ) V − ∗ ˜ f ν ,ν ∗ (Λ( z ) ∗ e ( z − z ) V − e ( z − z ) V ∗ Λ( z )) ∗ f ν ,ν o The gauge parameter Λ( z ) is given by (5.20) with Λ ( s ) = π ( z − z ) .We will explain the procedure by the case of ( ν , ν ) = (0 , V sm in Λ( z ) in terms of y a y b by using (7.2), and evaluating the trace in (7.38) for spin-stransformation, we have (cid:16) − i (cid:17) s − − π √ z ¯ z
12 2 s − X n =1 (2 s − n − s − n )! (cid:16) ( − i ) n − z n (cid:16) − iz (cid:17) s − n − ( i ) n − z n (cid:16) − iz (cid:17) s − n (cid:17) (7.39)Here z ij = z i − z j , etc . This summation can be performed by using the followingidentity. s X n =1 (2 s − n − s − n )! x n = (2 s − s − x − x F (cid:18) , , − s ; (cid:16) x − x (cid:17) (cid:19) + 12 ( − s ( s − (cid:16) x − x (cid:17) s (7.40)Here F is a hypergeometric function. Proof of this identity goes as follows. We27egin with a quadratic transformation of F .[44] F ( α, β, β ; x ) = (cid:18) − x (cid:19) − α F α , α + 12 , β + 12 ; (cid:18) x − x (cid:19) ! (7.41)In this formula we set α = 1 and β = 1 − s + ǫ , where ǫ is an infinitesimal parameter,which will be sent to 0. On the right hand side, we can safely take this limit, andobtain (2 / (2 − x )) F ( , , − s ; x / (2 − x ) ). On the left hand side, the secondand third parameter of the hypergeometric function become negative intergers inthis limit. By taking care of this point one obtainslim ǫ → F (1 , − s + ǫ, − s + 2 ǫ ; x )= ( s − s − s X n =1 (2 s − n − s − n )! x n − + 12 ( − s +1 [( s − (2 s − x s − (1 − x ) s . (7.42)This proves the formula (7.40).When this formula is applied to the summation of two terms in (7.39), thearguments of the two F ’s from both terms coincide and they cancel out. Only thesecond term in (7.40) contributes and the three-point function is given by < O (0 , ( z ) ¯ O (0 , ( z ) J ( s ) ( z ) > = g √ z ¯ z − s π ( s − z z z ! s . (7.43)Similarly the other three-point functions are calculated. All the results are summa-rized as follows. ( ν = 0 , < O (0 ,ν ) ( z ) ¯ O ( ν, ( z ) J ( s ) ( z ) > = 4 − s π ( s − z z z ! s < O (0 ,ν ) ( z ) ¯ O ( ν, ( z ) >, (7.44) < O (1 ,ν ) ( z ) ¯ O ( ν, ( z ) J ( s ) ( z ) > = 4 − s π ( s − s − z z z ! s < O (1 ,ν ) ( z ) ¯ O ( ν, ( z ) > . (7.45)Compared to eq (4.51) of [12] with λ = , our spin-s current J ( s ) is related to their J ( s )AKP as J ( s ) = ( − s +1 J ( s )AKP . Then the results for O (0 , and O (1 , agree.This result also shows that O (0 , , O (1 , and their partners ¯ O are primaries.28 Discussion
In this paper we studied a model of matter fields coupled to 3d higher-spingravity, where matter fields are a real 0-form C and a two-form B , and foundthat the solutions to the classical equations of motion of our theory also satisfythe linearized equations of motion of 3d Vasiliev gravity. A local boundary actionwhich yields two-point correlation functions in the boundary CFT is found, and amethod for calculating three-point functions for two primary operators and a spin-scurrent within the on-shell action method is presented. By using this method in HS [ ] × HS [ ] gravity, solutions to the equations of motion in AdS bcakgroundare found and the two-point functions of the primary operators dual to the matterfields are obtained. They agree with the results of linearized 3d Vasiliev gravity.The two-point functions of the fermion operators are also obtained.The interesting fact that the on-shell boundary term tr B C does not depend onthe coordinates is a consequence of the equations of motion for matter fields, i.e. ,covariantly-constancy conditions. As a result, our prescription for holography showsa novel feature: it is not necessary to take the near boundary limit. The holographicscreen is not at infinity. Data of the bulk is stored in the internal space sitting ata single point in the bulk. Due to the same reason, correlation functions of all theoperators with two scaling dimensions ∆ ± in ‘boundary CFT’ are obtained withoutdoubling the real matter fields C and B .It will be interesting to extend the present work also to the case of backgroundsother than AdS and BTZ black hole, such as higher-spin black hole geometry,and HS [ λ ] × HS [ λ ] gravity with arbitrary λ within 0 ≤ λ ≤
1. Finally, it is achallenging problem to introduce interactions among matter fields in an invariantway under higher-spin gauge transformations.
A Notations for sl(3,R) algebra
In this appendix notations related to sl (3 , R ) algebra are summarized.Let the generators L i ( i = − , , W n ( n = − , . . .
2) satisfy an sl (3 , R ) algebra.[ L i , L j ] = ( i − j ) L i + j , [ L i , W n ] = (2 i − n ) W i + n , [ W m , W n ] = −
13 ( m − n ) (2 m + 2 n − mn − L m + n (A.1)29e use the following representation of SL(3,R). L = , L = − , L − = − −
20 0 0 ,W = , W = − , W = 23 − ,W − = − , W − = (A.2)Non-vanishing norms of these matrices are given bytr ( L ) = 2 , tr ( L − L ) = − , tr ( W ) = 83 , tr ( W W − ) = − , tr ( W W − ) = 16 . (A.3) B Identities involving Moyal products
In addition to the formulae in appendix A of [11], the following results for Moyalproducts are useful for the calculation in this paper. F ( y ) ∗ e iy y = 0 ,F ( y ) ∗ e iy y = F (2 y ) e iy y ,e iy y ∗ F ( y ) = F (2 y ) e iy y ,e iy y ∗ F ( y ) = 0 , (B.1) e y T M y + ξ T y ∗ e y T N y + η T y = 1 p | L | exp n y T S y + 12 ξ T (1 + 2 X T N − X T σ − σ N XM ) y + 12 η T (1 + 2 XM + Xσ + σ M X T N ) y + η T X ξ − ξ T σ N X ξ + 12 η T σ M X T η o , (B.2)where M and N are symmetric matrices, and | L | = det M det N + tr( M σ N σ ) + 1, X = ( σ + M σ N ) − , S = σ XM + M X T N + N XM − σ X T N . Finally, e − i ( z − z ) y ∗ e − iy y ∗ e i (¯ z − ¯ z ) y ∗ e iy y = 12 ( z − z ) − (¯ z − ¯ z ) − exp (cid:16) iy y + iz − z y (cid:17) , (B.3)30 Scalar Two-point Functions on BTZ
The gauge fields (5.2) for BTZ black hole are given by [11] a = ( V + 14 τ V − ) dz, (C.1)¯ a = ( V − + 14¯ τ V ) d ¯ z. (C.2)Here τ and ¯ τ are modular parameters of the Eucllidean boundary torus. By using(3.37) and (3.38), solutions for C and B φρ are given by C ( ν ,ν ) = b − ∗ e − ( z − z ) ( V + τ V − ) ∗ ∗ f ν ,ν ( y ) ∗ e (¯ z − ¯ z ) ( V − + τ V ) ∗ ∗ b − , (C.3) B ( ν ,ν ) φρ = b ∗ e − (¯ z − ¯ z ) ( V − + τ V ) ∗ ∗ ˜ f ν ,ν ( y ) ∗ e ( z − z ) ( V + τ V − ) ∗ ∗ b. (C.4)Here b ( ρ ) = e ρ V ∗ and ν , ν = 0 ,
1. By substituting these solutions into the boundaryaction (6.1), two-point functions are obtained. h Ψ | O (0 , ( z , ¯ z ) ˜ O (0 , ( z , ¯ z ) | Ψ i = 14 g τ ¯ τ sin z τ sin ¯ z τ ! , (C.5) h Ψ | O (1 , ( z , ¯ z ) ˜ O (1 , ( z , ¯ z ) | Ψ i = 14 g τ ¯ τ sin z τ sin ¯ z τ ! , (C.6) h Ψ | O (0 , ( z , ¯ z ) ˜ O (1 , ( z , ¯ z ) | Ψ i = h Ψ | O (1 , ( z , ¯ z ) ˜ O (0 , ( z , ¯ z ) | Ψ i = 0(C.7)Here z = z − z , etc, and | Ψ i is an entangled state in a tensor product of twoCFTs. Both operators in these two-point functions live in the same Hilbert spaces H R or H L : h Ψ | O ( ν ,ν ) R,L ( z , ¯ z ) ˜ O ( ν ,ν ) R,L ( z , ¯ z ) | Ψ i . Two-point functions of a form h Ψ | O ( ν ,ν ) L ( z , ¯ z ) ˜ O ( ν ,ν ) R ( z , ¯ z ) | Ψ i , i.e. , mixed correlators, are obtained after a halfshift, z → πτ and ¯ z → π ¯ τ , in (C.5)-(C.7). These results agree with those in [11].Hence our prescription with the boundary term (6.1) also works for matter fieldson BTZ black hole. Furthermore, two-point functions of operators with differentscaling dimensions (C.7) vanish. References [1] E. S. Fradkin and M. A. Vasiliev,
On the gravitational interaction of masslesshigher spin fields , Phys. Lett. B 189 (1987) 89.[2] E. S. Fradkin and M. A. Vasiliev,
Candidate for the role of higher-spin gravity ,Ann. Phys. 177 (1987) 63. 313] M. A. Vasiliev,
Progress in higher spin gauge theories , [arXiv:hep-th/0104246].[4] M. P. Blencowe,
A consistent interacting massless higher-spin field theory inD=2+1 , Class. Quantum Grav. 6 (1989) 443-452.[5] M. R. Gaberdiel and R. Gopakumar,
An AdS dual for minimal model CFTs ,[arXiv:1011.2986 [hep-th]].[6] M. R. Gabardiel and T. Hartman, Symmetries of holographic minimal models ,[arXiv: 1101.2910 [hep-th]][7] C. Ahn,
The large N ’t Hooft limit of coset minimal models , [arXiv: 1106.0351[hep-th]].[8] M. R. Gaberdiel, R. Gopakumar, T. Hartman and S. Raju,
Partition functionsof holographic minimal models , [arXiv: 1106.1897 [hep-th]].[9] M. R. Gaberdiel and R. Gopakumar,
Minimal Model Holography ,[arXiv:hepth/1207.6697].[10] C-M. Chang and X. Yin,
Higher spin gravity with matter in AdS and its CFTdual , [arXiv:1106.2580[hep-th]].[11] P. Kraus and E. Perlmutter, Probing higher spin black holes , [arXiv:hep-th/1209.4937].[12] M. Ammon, P. Kraus and E. Perlmutter,
Scalar fields and three-point functionsin D = 3 higher spin gravity , [arXiv:hep-th/1111.3926].[13] E. Hijano, P. Kraus and E. Perlmutter, Matching four-point functions in higherspin AdS /CFT , [arXiv:hep-th/1302.6113].[14] C. Ahn, The higher spin currents in the N=1 stringy coset minimal model ,JHEP 04 (2013) 033, [arXiv:1211.2589 [hep-th]].[15] M. Henneaux and S.-J. Rey,
Nonlinear W ∞ as Asymptotic Symmetry ofThree-Dimensional Higher Spin Anti-de Sitter Gravity , JHEP 1012 (2010) 007,[arXiv:1008.4579 [hep-th]].[16] A. Campoleoni, S. Fredenagen, S. Pfenninger and S. Theisen, Asymp-totic symmetries of three-dimensional gravity coupled to higher-spin fields ,[arXiv:1008.4744 [hep-th]]. 3217] M. Gutperle and P. Kraus,
Higher spin black hole , [arXiv:1103.4304 [hep-th]].[18] M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter,
Spacetime geometry inhigher spin gravity , [arXiv:1106.4788 [hep-th]].[19] P. Kraus and E. Perlmutter,
Partition functions of higher spin black holes andtheir CFT duals , [arXiv:1108.2567 [hep-th]].[20] M. R. Gabardiel, T. Hartman and K. Jin,
Higher spin black holes fromCFT ,[arXiv:1203.0015 [hep-th]].[21] M. Ba˜nados, R. Canto and S. Theisen,
The action for higher spin black holesin three dimensions , [arXiv: 1204.5105 [hep-th]].[22] M. Ammon, M. Gutperle, P. Kraus and E. Perlmutter,
Black holes in threedimensional higher spin gravity: a review , [arXiv:1208.5182 [hepth]].[23] B. Chen, J. Long and Y-N. Wang,
Black holes in truncated Higher spin AdS3gravity , JHEP 1212 (2012) 052, [arXiv:1209.6185 [hep-th]].[24] B. Chen, J. Long and Y-N. Wang,
D2 Chern-Simons gravity , [arXiv:1211.6917[hep-th]].[25] C. Fronsdal,
Massless fields with integer spin , Phys. Rev.
D18 (1978) 3624; J.Fang and C. Fronsdal,
Massless fields with half-integer spin , Phys. Rev.
D18 (1978) 3630.[26] S. Lal and B. Sahoo,
Holographic renormalisation for the spin-3 theory and the(A)dS3/CFT2 correspondence , JHEP 1301 (2013) 004, [arXiv:1209.4804 [hep-th]].[27] A. Fotopoulos and M. Tsulaia,
Gauge invariant Lagrangians for free and in-teracting higher spin fields. A review of the BRST formulation , [arXiv:hep-th/0805.1346].[28] A. Campoleoni, S. Fredenhagen, S. Pfenninger and S. Theisen,
Towards metric-like higher-spin gauge theories in three dimensions , [ArXiv: 1208.1851 [hep-th]].[29] I. Fujisawa and R. Nakayama,
Second-Order Formalism for 3D Spin-3 Gravity ,Class. Quantum Grav. (2013) 035003, [ArXiv: 1209.0894 [hep-th]].3330] I. Fujisawa and R. Nakayama, Metric-Like Formalism for Matter Fields Coupledto 3D Higher Spin Gravity , Class. Quantum Grav. (2014) 015003, [ArXiv:1304.7941 [hep-th]].[31] A. Ach´ucarro and P. K. Townsend, A Chern-Simons action for three-dimensional anti-de Sitter supergravity theories , Phys. Lett. B180 (1986) 89.[32] E. Witten, , Nucl. Phys.B311 (1988) 46.[33] M. Henningson and K. Sfetsos,
Spinors and the AdS/CFT correspondence ,Phys. Lett.
B431 (1998) 63, arXiv:hep-th/9803251.[34] R.G. Leigh and M. Rozali,
The large N limit of the (2,0) superconformal fieldtheory , hep-th/9803068.[35] J. M. Maldacena,
The large N limit of superconformal field theories and super-gravity , Adv. Theor. Phys. (1998) 231, [arXiv:hep-th/9711200].[36] S. S. Gubser, I. R. Klebanov, A. M. Polyakov, Gauge theory cor-relators from non-critical string theory , Phys. Lett.
B428 (1998) 105,[arXiv:hep-th/9802109].[37] E. Witten,
Anti de Sitter space and holography , Adv. Theor. Math. Phys. (1998) 253, [arXiv:hep-th/9802150].[38] I. Klebanov and E. Witten, AdS/CFT correspondence and symmetry breaking ,Nucl. Phys B (1999) 89.[39] P. Breitenlohner and D.Z. Freedman,
Stability in gauged extended supergravity ,Ann. Phys. 144 (1982) 249.[40] S. F. Prokushkin and M. A. Vasiliev,
Higher spin gauge interactions for mas-sive matter fields in 3-D AdS space-time , Nucl. Phys. B (1999) 385[hep-th/9806236].[41] C. N. Pope, L. J. Romans and X. Shen,
W(infinity) and the Racah-Wigneralgebra , Nucl. Phys. B (1990) 191.[42] J. D. Brown and M. Henneaux, Commun. Math. Phys. (1986) 207.3443] M. Ba˜nados,
Three-dimensional quantum geometry and black holes ,arXiv:hep-th/9901148.[44] I.S. Gradshteyn and I. M. Ryzhyk,
Table of Integrals, Series, and Products,
Academic Press, 1994.[45] S. Giombi and X. Yin,
Higher spins in AdS and Twistorial Holography , JHEP , 086 (2011), [arXiv:1004.3736 [hep-th]].[46] S.J. Avis, C. J. Isham, and D. Storey,
Quantum field theory in anti-de Sitterspace-time , Phys. Rev.
D18 (1978) 3565.[47] M. Ba˜nados, C. Teitelboim, and J. Zanelli,
Black hole in three-dimensionalspacetime , Phys. Rev. Lett. (1992)69