Holographic Duality for 3D Spin-3 Gravity Coupled to Scalar Field

The 3d spin-3 gravity theory is holographically dual to a 2d ${\cal W}_3$-extended CFT. In a large-c limit the symmetry algebra of the CFT reduces to $SU(1,2) \times SU(1,2)$. On the ground of symmetry the dual bulk space-time will be given by an 8d group manifold $SU(1,2)$. Hence we need to introduce five extra coordinates in addition to three ordinary ones. The 3d space-time is a 3d hyper-surface $\Sigma$ embedded at constant values of the extra variables. Operators in the CFT at the boundary of $\Sigma$ are expressed in terms of ${\cal W}$ descendants of the operators at the boundary of $\Sigma_0$, where the extra variables vanish. In this paper it is shown that AdS/CFT correspondence for a scalar field coupled to 3d spin-3 gravity is realized in this auxiliary 8d space. A bulk-to-boundary propagator of a scalar field is found and a generating functional of boundary two-point functions of scalar ${\cal W}$-descendant operators is obtained by using the classical action for the scalar field. Classically, the scalar field must satisfy both Klein-Gordon equation and a third-order differential equation, which are related to the quadratic and cubic Casimir operators of $su(1,2)$. It is found that the coefficient function of the derivatives of the scalar field in the latter equation is the spin-3 gauge field, when restricted to the hypersurface. An action integral in the 8d auxiliary space for the 3d spin-3 gravity coupled to a scalar field is presented. An 8d local frame is introduced and the equations of motion for the 8d connections $A_{\mu}$, $\overline{A}_{\mu}$ are solved. By restricting those solutions onto $\Sigma$, flat connections in 3d $SL(3,\mathbb{R}) \times SL(3,\mathbb{R})$ Chern-Simons theory are obtained and new 3d black hole solutions with and without spin-3 charge are found by this method.


Introduction
After the discovery of AdS/CFT correspondence [1] this subject has been studied extensively and a lot of evidences have been accumulated until now. It is, however, still necessary to extend the range of applicability of this correspondence. One of those possible directions will be the 3d spin-3 gravity [2] [3] coupled to a matter (scalar) field. This is a 3d version of Vasiliev's theory [4] with higher spins consistently truncated up to 3. In this theory the gravity and spin-3 gauge field are described by SL(3, R) × SL(3, R) Chern-Simons gauge theory. However, the action integral for a scalar field which has spin-3 charge has not been found. Hence a check of duality for the correlation functions of conformal field theory (CFT) primary operators in the boundary W 3 conformal field theory (CFT) has not been carried out except for those of the currents. Therefore it is necessary to have a Lagrangian formulation of this coupled system. Study of the formulation of such a theory will also elucidate the nature of the 3d spin-3 gravity.
There is also another attempt to construct 2d CFTs which are dual to 3d higherspin gravity theories. [5] These 2d theories are based on 2d W N -minimal models which are obtained in terms of cosets of the form: This CFT has a central charge c N,k = (N − 1) 1 − N (N + 1) (N + k)(N + k + 1) ≤ (N − 1). (1.2) This model has W N algebra as a symmetry algebra. This is a special case of an extended symmetry algebra W ∞ [µ], which has all integer spins s ≥ 2 and which can be truncated to W N for µ = N . In the case of W 3 algebra N must be set to 3. The corresponding bulk theory is based on so-called hs[µ] algebra [6] and a scalar field with a suitable mass can be consistently coupled to the higher-spin gravity at the level of equations of motion.
In this paper the 3d spin-3 gravity theory [2] [3] which has a spin-3 gauge field in addition to a gravity field as well as a scalar field will be considered. On the boundary of AdS 3 space in the spin-3 gravity the CFT has an additional symmetry, W 3 symmetry.
This is a non-linear algebra. [7] [8] [ Actually, su(1, 2) is one of the two real forms of sl(3, C), distinct from sl(3, R). [3] 1 By combining the left and right sectors, the boundary field theory has global su(1, 2)×su (1,2) symmetry. Then by the principle of holography the bulk spin-3 gravity theory is also expected to have su(1, 2) × su(1, 2) asymptotic symmetry. 2 It was shown in [3] that the 3d higher-spin gravity theory can be formulated as a Chern-Simons gauge theory.
where k = ℓ AdS /4G and ℓ AdS is the AdS length 3 . The gauge group acts on the local frame fields, A = ω + e and A = ω − e, and in the case of the spin-N gravity the gauge group 1 Conventions for sl(3, R) and su (1,2) used in this paper are given in appendices A and B. The difference of the two is the sign of the right hand side of the commutators [Wm, Wn]. 2 In [15] it was assumed that the symmetry of the spin-3 gravity is sl(3, R) × sl(3, R)). The symmetry of the bulk space-time is, however, su(1, 2)×su (1,2), while sl(3, R)×sl(3, R) is a symmetry of the local frame. Translation from sl(3, R) to su(1, 2) is simply carried out by analytic continuation of some variables. See footnote 7. 3 In this paper ℓ AdS will be sometimes set to 1. is SL(N, R) × SL(N, R). The gauge connections must satisfy suitable conditions on the boundary in order for the boundary value problem to be well-posed. The usually adopted boundary condition is A − = A + = 0. In [9][10] an asymptotically AdS 3 solution which shows UV/IR interpolating behavior were found and a black hole solution with spin-3 charge was also obtained. In these solutions all components A ± and A ± do not vanish and the boundary conditions are imposed on the components of connections A ± = 8 a=1 A a ± t a separately, where t a is a generator of sl (3, R), and the W 3 algebra of the higher-spin currents in the CFT is realized as Ward identities of the currents in the presence of a perturbation term d 2 zµW in the action integral. See [27].
When matter fields such as scalar fields are coupled to higher-spin gravity, it is still possible to describe matter degrees of freedom by means of Wilson lines. [11] [12] However, a natural description of matter degrees of freedom and their coupling to spin-3 gravity in terms of Lagrangian formalism are still missing. For matter fields which have spin-3 charges it is not possible to write down an action which is invariant under spin-3 gauge transformations as well as diffeomorphisms. It is not possible to derive boundary conformal field theory (CFT) correlation functions from on-shell action by using the standard differentiating dictionary of holography, either. The purpose of this paper is to improve this situation. The action integral for scalar fields and spin-3 gravity fields are presented in a higher-dimensional setting, which will be explained below. It is shown that when a solution to an equation of motion for a scalar field is substituted into the scalar action, a generating functional for a two-point function of scalar operators is obtained semiclassically. The source functions work as boundary conditions of the scalar fields.
There is another motivation for the present work. When the symmetry of the 2d CFT is W-extended, all states of a scalar field must be reconstructed [13] [14] in the bulk from W-descendants of a scalar primary state in the boundary CFT. In this paper we will introduce an 8d auxiliary space dual to W 3 CFT by following our previous paper [15].
Usually in 2d (Euclidean) CFT without W 3 symmetry, a representation of global Virasoro algebra is constructed as follows. In a highest-weight representation a highest-weight state |h which satisfies L h 0 |h = h |h and L h 1 |h = 0 is introduced. Here the generators with a superscript h represent those in the hyperbolic representation [17], which is appropriate for Lorentzian Poincaré coordinates. These are defined in (2.13) and (2.18) of [15]. L h ±1 and L h 0 are global Virasoro generators. Then any states in this representation are given by linear combinations of descendants (L h −1 ) n |h (n = 0, 1, 2, . . .). These states are combined into a single state |φ(x) = exp{ixL h −1 }|h by introducing a coordinate x. A shift in x corresponds to a translation. In the case of a global large-c W 3 algebra highest-weight representation is defined by a highest-weight state|h, q , which satisfies L h 0 |h, q = h|h, q , W h 0 |h, q = q|h, q , and L h 1 |h, q = W h 1 |h, q = W h 2 |h, q = 0. 4 Any descendant states in this representation are linear combinations of states of a form By introducing variables x, α and β, these states are combined into a single state |φ(x, α, β) = exp{ixL h −1 } exp{−αW h −2 } exp{iβW h −1 } |h, q . By using the coefficients of the Taylor expansion of this state in the variables, x, α, β, any states can be obtained. Due to the left and right movers it turns out it is necessary to introduce six variables to describe states in W 3 extended CFT. 5 As for the variables in the bulk it is necessary to additionally introduce a radial coordinate y (y = 0 is the boundary), which corresponds to L h 0 + L h 0 , and another coordinate γ, corresponding to Hence the 'bulk space-time' holographically dual to the boundary CFT with W 3 symmetry is 8 dimensional.
In the remainder of this section we will give a review of our paper [15]. At the end of this section the content of this paper will be presented. In [15] we constructed a state in the boundary W 3 CFT which represents the one of a scalar field put at one point inside the bulk. In the case of ordinary AdS 3 /CFT 2 without spin-3 gauge field, such a state |Φ(y = 1, x + = 0, x − = 0) at the center of AdS 3 (in the Poincaré coordinates) must satisfy sl(2, R) conditions. [16][17] [18] [19] By the action of exp{ix h 0 a state at any point in the bulk is obtained. 6 In the case of spin-3 gravity, a state of a scalar field |Φ(x + , α + , β + , x − , α − , β − , y, γ) must satisfy su(1, 2) conditions. 7  In [15] the state of a scalar field in the boundary W 3 CFT was explicitly constructed, and although this state is a formal integral expression, the existence of such a state was established. Then by using exponentials of W 3 generators the state for a scalar field at any point in the bulk was also obtained. During this work an infinite-dimensional representation of W 3 generators in the bulk in terms of differential operators were also 4 In [15] W3-charge was denoted as µ. In this paper it will be denoted as q instead. Later µ is used for a chemical potential. 5 Wakimoto representation of large-c W3 algebra is expressed in terms of similar variables, whose relation to x, α, β is not known, and this representation was used for calculating correlation functions in W3 extended CFT in [22]. 6 Light-cone coordinates are defined by x ± = t ± x. In the case of black holes in sec.4 x ± = t ± φ. 7 In [15] we solved an sl(3, R) conditions for the state in the bulk. To convert the results of [15] to those appropriate for su (1,2), it is necessary to make substitutions, obtained. This is presented in Appendix B, because some variables are redefined compared to those in [15] by analytic continuation.
From the structure of the local state in the bulk it was also found that the scalar local state in the bulk satisfies a partial differential equation which is associated with the quadratic Casimir operator of su(1, 2) 8 .
By using 9 the representation (B.1)-(B.2) and where |O ∆,q is a CFT primary state on the boundary with eigenvalues L h 0 =L h 0 = ∆/2 and W h 0 = W h 0 = q, a differential equation for a scalar field in 8d space was derived. It takes a form where ∇ 2 is a Laplacian in a 8d space, which has a metric (1.15) defined below. m is a mass of the scalar field related to ∆ and q by The explicit form of the equation is presented in Appendix C. Conversely, ∆ is given in terms of the mass and charge as ∆ = 4 + m 2 + 16 + 3q 2 . (1.14) In [15] this equation was interpreted as a Klein-Gordon equation for a scalar field in the 8d space. Because the scalar field transforms non-trivially under SU (1, 2) × SU (1, 2), its equation of motion must be formulated in 8d. The equation (C.1) coincides with Klein-Gordon equation for a scalar field in a space-time with a metric: 8 See Appendix A and B for conventions. 9 In replacing sl(3, R) with su(1, 2) the eigenvalue µ of W0 must also be replaced by −iq.
• For β + β − < 0 the space-time on the hypersurface is not AdS, but asymptotically AdS. This is a solution interpolating two vacua: one corresponding to UV CFT at y = 0 with a AdS length ℓ ′ AdS = 1 2 ℓ AdS = 1 2 , and the other to IR CFT at y = ∞ with AdS length ℓ AdS = 1. Hence conformal symmetry is broken in the boundary field theory for non-zero β + β − .
It was observed in [15] that the parameters β ± play the role of flow parameters of renormalization group. To identify this flow in the bulk let us set β + = −β − ≡ β in (1. 16) for simplicity.
In general, in the bulk of an asymptotically AdS space-time AdS symmetry is broken at y = 0 and conformal symmetry is also broken on the holographic screen located at this value of y. As y gets closer to 0, then AdS symmetry will be recovered and the field theory on the holographic screen will flow in the UV to a fixed point, if it exists. Now, to describe a new type of flow on the holographic screen, we should change the radial variable y to a new one z as y = 2 √ βz. Then the metric (1.17) is transformed to If y and γ are fixed, this is an asymptotically AdS metric, where z is a new radial coordinate. In this case we can consider a flow on a constant-y holographic screen by fixing the value of y and sending z → 0. Then the field theory flows to a UV fixed point and the above metric flows to that of AdS 3 with AdS length ℓ ′ AdS . Along the way β goes to ∞. This flow is depicted in Fig.1. On the CFT side this flow is associated with a change of the translation operator L h −1 → −(1/4)W h −2 as follows. [15] On the boundary of a chosen hypersurface Σ αβγ (holomorphic) primary operators in general have a form, Let us concentrate on (global) Virasoro symmetry, because α, β and γ are fixed. For simplicity we set γ = 0 here. When the correlation functions of scalar operators on the common boundary of Σ αβγ are computed, they depend on β + in addition to x, because the exponentials exp{ixL h −1 } and exp{βW h −1 } do not commute. They do not depend on α + , because [L h −1 , W h −2 ] = 0. For simplicity α + will be omitted in the following discussion. Then an operator O(x + , β + ) on the boundary are rewritten as where (1. 22) In the limit β + → ∞ (1.22) asymptotes to −β + W h −2 , which is proportional to the Virasoro [20]. And the central charge of W In this paper we will show that holography of 3d spin-3 gravity and the boundary W 3 CFT is realized in the auxiliary 8d space. This will be done in the following steps. In the above discussion Klein-Gordon equation is associated with the quadratic Casimir operator.
In sec. 2 of this paper the other equation for the scalar field, which stems from the cubic Casimir operator of su(1, 2), is studied. It is shown that the coefficient functions of the third-order derivatives in this third-order differential equation coincide with the spin-3 gauge field ϕ µνλ , when restricted to 3d hypersurface Σ αβγ . In sec. 3 a bulk to boundary propagator is obtained by solving the Klein-Gordon equation and it is shown that semiclassically evaluated path integral for a free scalar field in the background (1.15) yields a generating functional for a two-point function of scalar operators on the boundary. In sec.
4 an 8d action integral for the spin-3 gravity coupled to a scalar field is proposed. In sec. properties A x − = 0 and A x + = 0. The integrability condition for the partition function of the charged black hole is checked. The partition function coincides with that of the solution obtained in [9], although the flat connections of the two solutions satisfy distinct boundary conditions. This paper is summarized in Sec. 6. In appendix A conventions for sl(3, R) and su(1, 2) algebras in this paper are presented. In appendix B a representation of W 3 generators in terms of differential operators is presented. In appendices C and D the explicit forms of Klein-Gordon equation for a scalar field Φ and a spin-3 field φ µνλ in the auxiliary space is presented. Black hole solutions are obtained by adding extra terms ψ and ψ to the flat connections for su(1, 2) × su(1, 2) symmetric space-time. In appendix E equations for their first-order perturbations ψ (1) and ψ (1) are presented and the solutions to them are shown. In appendices F, G and H results for black hole solutions with and without spin-3 charge are presented.

Equation Related to the Cubic Casimir Operator
As explained in sec. 1 the local state for a scalar field in the bulk |Φ( satisfies an eigenstate equation for the quadratic Casimir operator. This state also satisfies an equation corresponding to the cubic Casimir: where h = ∆/2 and q are the conformal weight and the spin-3 charge of the boundary primary state |O ∆,q .C 3 is defined in (A.14). By substituting (B.1), (B.2) into (2.1) a differential equation for a scalar field Φ in the bulk is obtained. After some calculation, this can be succinctly written in the following simple form.
where ∇ µ is a covariant derivative for the metric (1.15) with the Christoffel symbol, and φ µνλ is a completely symmetric tensor. The explicit form of this field is given in Appendix D. The imaginary unit i on the right hand side of (2.2) implies that Φ is a complex function. (See the solution (3.7).) On a 3d hypersurface Σ αβγ with constant α ± , β ± and γ, which was introduced in sec This field breaks Lorentz symmetry on the boundary of Σ αβγ , on which the values of β ± are fixed. It is also found that 8d covariant derivative of φ µνλ vanishes.
This fact gives a geometrical meaning to the φ µνλ field.
It will soon be shown that φ µνλ actually coincides with a spin-3 gauge field on a hypersurface Σ αβγ . This suggests that both Casimir equations have geometrical meaning via the metric and spin-3 gauge field. For this purpose we introduce a vielbein field e a µ . Here a = 1, 2, . . . , 8 and µ = x ± , α ± , β ± , y, γ. This is a local-frame field in 8 dimensions and is an 8 × 8 matrix. Notice that this is different from the 3 × 8 rectangular vielbein introduced in [3]. It is required that 11 To impose more restrictions we consider sl(3, R) connections 12 and require flatness conditions on them.
Note that these are equations in 8d, and there are no Chern-Simons actions which classically lead to (2.8) and (2.9). 13 On a hypersurface Σ αβγ with constant α ± , β ± and γ, however, these frame fields reduce to 3 × 8 rectangular matrices, and these equations are nothing but the equations of motion for connections in SL(3, R) × SL(3, R) Chern-Simons gauge theory.
It turns out there are two distinct solutions to (2.8) and (2.9), which produce (1.15).
• Solution (I) Throughout this paper the vielbein e a µ is assumed to be invertible. 12 See appendix A for our conventions for sl(3, R) algebra. The reason for using sl(3, R) generators ta, not su(1, 2) onesta is that if su(1, 2) generators are used, then it turns out that e a µ (a = 4 ∼ 8) becomes pure imaginary. If sl(3, R) generators are used instead, e a µ is real. Hence the symmetry algebra of the local frame is sl(3, R) × sl(3, R), while that of the space-time is su(1, 2) × su(1, 2). The vielbein connects two analytically continued spaces. 13 However, as will be discussed later, there exists an 8d Einstein-like action such that its equations of motion coincide with (2.8) and (2.9), provided the metricity condition and invertibility of the vielbein are assumed for the local frame fields.
We note that both A x + and A x − are non-vanishing. Similarly A x ± = 0. As β ± increase from 0 to ∞ in this solution, the leading terms interchange between A x + and A x − . When this solution is restricted to a hypersurface Σ αβγ (especially for γ = 0), it coincides with the interpolating solution eq (2.27) of [9]. On the other hand in the case of solution (I), although the metric agrees with (1.15), the spin-3 gauge field (2.14) does not coincide with (D.1). For simplicity, only the result for ϕ on a hypersurface Σ αβγ is presented here.
On Σ 0 two solutions (I) and (II) coincide. Moving away from Σ 0 , they do not agree. 2) for a scalar field coincides with the spin-3 gauge field (2.14). This is in accord with the fact that the coefficient function g µν of derivatives in the Klein-Gordon equation is the metric field. The two equations for the scalar field are written in terms of the geometrical quantities. On the boundary of Σ αβγ this solution satisfies boundary conditions A x − = A x + = 0. The connection (II) on Σ αβγ is the interpolating solution between IR and UV.
In sec. 5 we will construct black hole solutions by extending this solution.
3 Bulk-to-Boundary Propagator In this section a bulk-to-boundary propagator for a scalar field Φ propagating in the .
where D 12 , etc are defined by q is obtained by power series expansion in y near the boundary y ∼ 0 14 : due to the boundary condition. The exponential factor for the bulk point is introduced to the first term in (3.3). By substituting this solution into (C.1) an equation for f 1 is obtained and it is readily solved. By repeating this procedure we get the following Now up to order y ∆+4 the series (3.3) can be summed up with the following result.
It is directly checked that (3.7) solves Klein-Gordon equation (C.1) exactly. Furthermore, it is checked that this propagator also satisfies up to order y ∆+4 the equation (2.2) which is related to the cubic Casimir operator. As an independent check, we also found that 2), which act on the i-th variable. Hence it is established that this bulk-to-boundary propagator is an exact solution.
By using the bulk-to-boundary propagator a scalar field inside the bulk is reconstructed in terms of a boundary CFT operator. This provides a more explicit expression for the extrapolating dictionary than that of the local state for a scalar field obtained in eq (3.9) of [15].
All W -descendants of the primary scalar operators correspond to the scalar field in the bulk. Now let us switch to a Euclidean space by a Wick rotation.
Herez,ξ andζ are complex conjugates of z, ξ, ζ. The metric (1.15) becomes It can be shown that in the region including z 12 , ξ 12 , ζ 12 ∼ 0 and in the y → 0 limit K ∆,q behaves as Here the dots stand for terms with higher order powers of y, and D E 12 , . . . are obtained by replacing variables in D 12 , . . . according to the rule of analytic continuation (3.9). N (γ) is a function of γ: We define action integral for a scalar field coupled to spin-3 gravity by When a solution to the Klein-Gordon equation where φ(z, ξ, ζ) is a boundary condition, is substituted into (3.13) and is used, a generating functional for the two-point function is obtained as a surface integral on the boundary by using the standard method [23]. 15 Hence both holographic dictionaries of AdS/CFT, the WGKP[23] [24] and BDHM dictionaries [13], also hold in the case of 3d spin-3 gravity which couples to a scalar field.

Action Integral for 3D Spin-3 Gravity Coupled to a Scalar Field
One of the purpose of this paper is to find out a formulation of 3d spin-3 gravity coupled to a scalar field in terms of 8d auxiliary bulk space-time. A natural action integral for the scalar field (3.13) was found in sec. 3. To make this formulation complete, it is necessary to write down an action for the gravity sector in the 8d auxiliary space-time.
In (2.6) and (2.7) we introduced 8d vielbein e a µ , spin connection ω a µ and gauge connections A = ω + e and A = ω − e. By solving flatness conditions for the connections de a + f a bc ω b ∧ e c = 0, (4.1) These are linear combinations of the flatness conditions. In terms of components the first equation is given by a torsionless condition Here ∇ µ is a covariant derivative for g µν = e a µ e aν . If we restrict solutions to (4.3) to satisfy the vielbein postulate, which states that the full covariant derivative of e a µ should vanish, 16 then by using (A.9) the spin connection ω a µ is expressed in terms of e a µ , We define the field strength for the gauge field of local frame sl(3, R) transformation.
The second equation of the flatness condition (4.2) is now written as This equation can be derived from the following action. This will not be attempted in this paper. On the hypersurface with constant α ± , β ± , γ, solutions to the equations of motion for (4.8) reduce to those in 3d spin-3 gravity represented by SL(3, R) × SL(3, R) Chern-Simons theory.
Action for a free charged scalar field is given by to yield a set of Casimir operators, the quadratic-order differential equation will be also sufficient to determine the solution. The complex scalar field Φ is assumed to have spin-3 charge q. Then the scaling dimension ∆ of Φ is determined by (1.13). The self coupling of scalar fields can be introduced straightforwardly. We propose that the total action S total = S spin-3 gravity + S scalar (4.10) describes the 3d spin-3 gravity theory coupled to a scalar field. In the limit of large central charge (G → 0), the action for spin-3 gravity determines the geometry of the 8d space semi-classically, and S scalar describes the scalar field in this background.

Black Hole Solutions
In this section perturbations around the background solution (2.12) and (2.13) are considered and new solutions to the flatness conditions, black hole solutions, are obtained.
First we will consider flat connections A (2.12) and A (2.13) at y = 1, which will be denoted as A 0 and A 0 , respectively.

1)
Gauge connections at an arbitrary value of y are obtained by carrying out the following gauge transformations 3) where b(y) = y t 2 . A 0 and A 0 also satisfy the flatness conditions.
Now we add small perturbations ψ and ψ to A 0 and A 0 , respectively: Then we impose conditions that A and A should satisfy the flatness conditions (5.5), (5.6).
Finally, gauge transformations b(y) are used to obtain connections A and A.
When ψ and ψ are expanded as ψ = ψ (1) + ψ (2) + · · · and similarly for ψ, where ψ (i) is an infinitesimal one-form at i-th order of perturbation, the flatness conditions to first order read These conditions will be solved explicitly. By expanding ψ (1) and A 0 into a basis of sl(3, R) generators as where summation over a = 1, · · · , 8 is not shown explicitly, eq (5.9) is transformed into

Asymptotically AdS 3 Black Hole Solutions without Spin-3 Charge
The flat connections for a black hole without spin-3 charge are obtained by choosing suitable Q n and Q n , which yield static or stationary connections. The exact terms dQ n , dQ n in (E.2), (E.3) are determined to make ψ (1)a and ψ (1)a periodic in x ± . The results for Q n are presented in (F.1). The results for ψ (1)a and ψ (1)a are also presented in (F. 2) in appendix F. In these results parameters a andā are the following constants. On a hypersurface Σ αβγ with constant α ± , β ± , γ, the metric (F.5) reduces to The induced metric (5.17) is also a solution to the equation of motion of 3d SL(3, R) ×

SL(3, R) Chern-Simons gauge theory. So this is a new black hole solution in the 3d
space-time. This black hole does not have spin-3 charge. On a hypersurface Σ 0 with α = β = γ = 0 this metric coincides with that of BTZ black hole [25]. This metric changes from one Σ αβγ to another Σ ′ αβγ , when the values of β ± , α ± , γ are changed. If β ± = 0, the leading behavior of the metric near y ∼ 0 is y −4 and the space-time is asymptotically AdS with AdS length = 1/2. As for the spin-3 field we checked that as in (2.4) ϕ for these flat connections A, A satisfy the 8d equation, Result for the spin-3 field will not be presented here, because it is complicated. On the hypersurface Σ 0 , where α ± = β ± = γ = 0, it vanishes.
Hence the hypersuface Σ 0 is exactly the BTZ black hole. On other Σ's spin-3 field ϕ does On the hypersurface Σ αβγ this reduces to A x + = U −1 ∂ x + U 17 and U is solved as U = exp x + A x + . On the 3d Euclidean asymptotically AdS space, which is obtained by Wick rotation from Σ, the coordinates x + = x + it E ≡ z and x − = x − it E ≡ −z are identified as (z,z) ∼ (z + 2πτ,z + 2πτ ), where τ andτ are modular parameters of the boundary tori. A holonomy matrix w is defined by U (z,z) −1 U (z + 2πτ,z + 2πτ ) = exp w. Hence w is given by This is computed by using (5.7), (5.1) and (F.2). Similarly A = A 0 + ψ definesw.
By requiring that the flat connections are non-singular, the matrices w,w should be required to have the same eigenvalues as those for the vacuum. Hence they need to satisfy 17 Notice that A = A x + dx + on the hypersurface and A x + does not depend on x + . the conditions, [9] det w = 0, (5.23) tr w 2 = −8π 2 , (5.24) and similar equations forw. It can be shown that the first condition is trivially satisfied.
The second one yields Since τ is related to the inverse right and left temperatures, β R and β L , as τ = i 2π β R and τ = i 2π β L , respectively, we obtain where M and J are mass and angular momentum of the black hole. Similarly, for the left inverse temperature we have Hence α ± , β ± and γ do not appear in the temperatures. Now let us investigate whether the β + = −β − ≡ λ → ∞ limit of the metric (5.17) exists. Some calculation shows that even if coordinates y, x ± are rescaled, 18  However, if a andā are also rescaled in an appropriate way, finite limits exist. Because this may produce new solutions, we will study such limits. We perform the following rescaling of variables in (5.17), 19 as well as while α ± and γ are fixed. The metric has a finite limit for the constant ρ ≥ 1. The limit depends on (1) ρ = 1 or (2) ρ > 1. 18 In the case of the metric (1.16) y must also be rescaled as y = λ 1/2ỹ in order to take a finite limit of the metric as λ → ∞, where β + = −β − = λ. 19 Here the AdS length is set to 1. The metric on the hypersurface Σ αβγ is a solution to the equations of motion of 3d SL(3, R) × SL(3, R) Chern-Simons theory for each value of α ± , β ± and γ. On Σ αβγ , β ± are not coordinates, but just constants. Hence the constants a,ā and other variables can be rescaled and made dependent on λ.
(1) ρ = 1: In the limit λ → ∞, the 3d metric (5.17) asymptotes to If (a−1)(ā−1) > 0, the signature of the metric is correct and this is an asymptotically AdS black hole solution with AdS length ℓ ′ AdS = 1 2 . If (a − 1)(ā − 1) = 0, this is an asymptotically AdS black hole with the AdS length ℓ AdS = 1. Metric (5.30) is a solution to the equations of motion in the spin-3 gravity based on the 3d Chern-Simons theory. The 3d metric (5.30) depends on the parameter γ in addition to a andā, the mass and angular momentum. As for the spin-3 field, it also has a well-defined λ → ∞ limit.
(2) ρ > 1: In the limit λ → ∞, the 3d metric (5.17) asymptotes to This coincides with the metric at α ± = β ± = 0 There are terms which contain γ in (5.32). Due to the factor cosh γ this is a deformed BTZ solution. As for the spin-3 field, it also has a well-defined λ → ∞ limit.

Black Hole Solution with Spin-3 Charges
In this subsection the functions Q n , Q n and connections ψ, ψ for the black hole solution with spin-3 charge will be constructed. This is more difficult than the preceding black hole, because more parameters than the mass and angular momentum must be introduced and the integrability condition for the partition function needs to be taken into account [9].
The result is presented in (G.1), (G.2) in appendix G. The result includes parameters b,b, µ,μ, which are spin-3 charges and chemical potentials, in addition to a andā. ψ (1) and ψ (1) are contributions to first order of these parameters. When the black hole has spin-3 charges, however, ψ and ψ do not stop at the first order. It is then necessary to expand ψ as ψ = ψ (1) + ψ (2) + · · · and systematically solve the equation for ψ (i) . The i-th-order perturbation ψ (i) needs to satisfy Suppose that ψ (k) for k = 1, 2, · · · , i − 1 has been obtained. Then the right hand side of n at the second and third orders of perturbation, which will yield new contributions to ψ (2)a and ψ (3)a according to (E.2).
It is also necessary to introduce other new terms to ψ (2)a , which are proportional to the second order infinitesimal parameters, µa, µb, but otherwise must have been included in the first order connection ψ (1)a . The results for ψ (2) and ψ (2) are given in (G.3). We carried out analysis to the fourth order and the results for ψ (3) are presented in (G.4). These solutions contain extra constant parameters ζ i ,ζ i (i = 1, 2, 3), which will be determined by the condition of integrability shortly. As will be clear from the result, the solution ψ does not have -components, while ψ does not have + components. The perturbation expansions do not seem to terminate at a finite order.
The metric of the black hole solution with spin-3 charge up to the first order in b,b, µ,μ is given by where ds 2 0 is given in (1.15) and the other terms are presented in (5.35). On Σ 0 the black hole metric does not coincide with that of the black hole solution with spin-3 charge obtained in [9].
Here ν is defined by ν = 4µτ. We need to setν = 4μτ . Then the left-mover temperature is given by Entropy of the black hole S = S R + S L can be obtained by the method used in [9]. The right-moving part S R is given by S R = (πℓ AdS /2G) √ af (27b 2 /2a 3 ), where f (y) = cos θ, θ = arctan[ y(2 − y)/6(1 − y)]. The entropy and partition function do not depend on α and β. A scalar field operator in this black hole background will be dual to all Wdescendants of some scalar operator in a W 3 -extended CFT at finite temperature.

Black Hole Solution with Spin-3 Charge on Σ 0
As mentioned above, although the flat connections are already complicated even at the fourth order of perturbation, miraculous cancellation occurs in the holonomy conditions, (5.38) and (5.39). The variables α ± , β ± and γ corresponding to Σ αβγ do not appear in these conditions. This situation is similar to that in the black hole solution without spin-3 charge, which was observed in subsec. 5.1. So, let us study the holonomy conditions for the flat connections on the hypersurface Σ 0 , where α ± = β ± = γ = 0. From the results in appendix D, after setting α ± = β ± = γ = 0 the flat connections on Σ 0 are given by  conditions. Hence this may support the expectation that the conditions (5.38) and (5.39) will remain valid, even if higher-order terms which depend on α, β, γ are included in ψ and ψ. From (5.46)-(5.47) the metric is obtained as follows.
General black hole solutions with spin-3 charge will be obtained by making a, b and µ in A| Σ 0 depend on x + . Similarly,ā,b andμ in A| Σ 0 are replaced by functions of x − .
These connections are still flat. These connections are further required to satisfy the holonomy conditions det w = 0 and tr w 2 = −8π 2 , which are similar to (5.38) and (5.39) but more complicated. Here the holonomy matrix w is defined as before now by using a path-ordered exponential U (x + ) = P exp{− It is known that when b = µ =b =μ = 0, the metric constructed from these connections is the most general BTZ metric in the Fefferman-Graham gauge [25]. When A is transformed as A → U −1 AU + U −1 dU with U = exp t a λ a (x + ) by restricting infinitesimal parameters λ a to keep the form of A and imposing δµ = 0, transformations δa and δb are obtained. For µ = 0 these transformations generate the W 3 algebra. [3] Next we consider connections which do not satisfy A − , A + = 0.
It can be shown that these connections are flat and up to change of notations these coincide with eq (4.1) of [10]. Then the metric derived from these connections is given as follows and does not coincide with (5.48).
ds ′2 = 1 y 2 dy 2 + 1 3 3a(1 + 8bμy 2 ) + 4ā 2μ (3by 4 + 4μ) (dx + ) 2 It is checked that the spin-3 field obtained from ( This means that the partition functions of the would-be CFT's dual to each backgrounds will coincide. These 8d flat connections might be related by large gauge transformations. Let us study the boundary conditions for the connections. The variation of the CS action (1.7) is given by To make this vanish after the bulk equation of motion is used, δA ′ must satisfy tr (A ′ + δA ′ − − A ′ − δA ′ + ) = 0 on the boundary. Usually, the boundary condition A ′a − = 0 or A ′a + = 0 is imposed as (5.49). The connection (5.50), however, satisfies the following conditions, one for each canonical pair, where A ′ is obtained from A ′ by the transformation (5.3). These determine the Dirichlet conditions. Then a variation of the action vanishes after addition of extra local terms (k/4π) y=ǫ (4y −1 A ′1 − + 16µ y −2 A ′4 + ) d 2 x to the action (1.7). Here ǫ is a UV cutoff and finally a limit ǫ → 0 must be taken. Under this variation µ should not be changed. Hence there exist appropriate boundary conditions.
Remaining problem is how to compute partition functions in an explicit manner. It can also be shown that the metric (5.48) is in a wormhole gauge as the black hole solution in [9]. Note, however, that in the limit a, b, µ → 0 only the flat connections (5.46)- (5.47) correspond to the 8d vielbein which reproduces the spin-3 field (D.1) and the coefficient function of the cubic equation for the scalar field.

Summary and Discussions
In this paper a formulation of 3d spin-3 gravity coupled to a scalar field is studied from the view point of a realization of W 3 symmetry in the bulk space-time, not in the local frame. It is shown that this formulation is possible in the extended 8d space. In the most symmetric case this is a group manifold SU (1,2). In this 8d space holographic duality between the bulk and the boundary is explicitly realized. The ordinary 3d bulk is obtained is found to coincide with that of the black hole solution with different boundary condition obtained before in [9]. Further investigation of the black hole solution (5.46)-(5.47) is necessary.
In sec. 1 it is shown that the scaling dimension ∆ of a scalar operator on the boundary is related to the scalar mass m and spin-3 charge q by (1.14). This is similar to the ordinary dictionary for the simple AdS 3 gravity, ∆ = 1 + √ 1 + m 2 , but slightly modified. In the case of 3d higher-spin gauge theory dual to W N minimal model it was shown that a scalar Riemann tensor can be expressed in terms of Ricci tensor. Is there a similar identity in spin-3 gravity? In this paper a bulk-to-boundary propagator for a scalar field is calculated.
If the bulk-to-bulk propagator is obtained, conformal blocks of W 3 extended CFT may be studied by using the methods of holography.
Finally, we had to introduce 8d space-time, which is a deformation of SU (1, 2), to realize holographic duality of W 3 CFT and spin-3 gravity. In [31]

sl(3, R) Algebra
Generators of sℓ (3, R) in the fundamental representation are given [3] by The structure constants f ab c are defined by and a Killing metric h ab is given by Casimir operators are given by where T a is some irreducible representation of sl (3, R). The matricest a generate the W 3 wedge algebra (1.5) after an identificationt 1 = L 1 , Killing metrich ab = (1/2) tr (t atb ) is given byh 22

B Representation of W 3 Generators by Differential Operators
Here the infinite-dimensional representation of su(1, 2) × su(1, 2) algebra in the hyperbolic representation [17] is presented. This is a representation for the generators of transformations in the bulk. This is obtained from eqs (4.3) and (4.4) of [15] by replace- These generators satisfy the algebra (1.5). Generators L h n , W h n are obtained from the above by interchanges,

C Klein-Gordon Equation for a Scalar Field in the 8d Spacetime
Here an explicit form of the Klein-Gordon equation for |Φ , (1.12), is presented.
Similarly, general solution to (5.10) is given by Hence there are 16 perturbative modes Q n , Q n in the classical solutions. Actually, these are gauge modes. However, when those modes are changed by amounts which are not single-valued functions on the torus, then the flat connections before and after the change are inequivalent. For static or stationary black hole solutions the functions Q n and Q n have to be chosen such that ψ (1)a and ψ (1)a are periodic in x ± .

F Black Hole Solution without Spin-3 Charge
The flat connections for a black hole without spin-3 charge are obtained by choosing suitable Q n and Q n in (E.2) and (E.3), which yield static or stationary connections.
For simplicity the variable γ is not included in Q n . Q n is obtained by the following replacement.

G Black Hole Solution with Spin-3 Charge up to Third Order
Functions Q n and Q n for static or stationary black holes with spin-3 charge are given as follows. It turned out it is necessary to include higher order corrections to ψ a and ψ a .
Higher-order corrections ψ (i)a , ψ (i)a are also presented.
Q n is obtained by the following replacement.
Parameters a,ā are related to the mass and the angular momentum as in (5.14), (5.15), and b andb to the spin-3 charges. µ andμ are chemical potentials for the charges. The result for the spin-3 gauge field is not presented. It is also checked that ϕ satisfies the equation, ∇ µ ϕ νλρ = 0.