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The 3d spin-3 gravity theory is holographically dual to a 2d ${\calW}_3$-extended CFT. In a large-c limit the symmetry algebra of the CFT reducesto $SU(1,2) \times SU(1,2)$. On the ground of symmetry the dual bulk space-timewill be given by an 8d group manifold $SU(1,2)$. Hence we need to introducefive extra coordinates in addition to three ordinary ones. The 3d space-time isa 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 $\Sigma_0$, wherethe extra variables vanish. In this paper it is shown that AdS/CFTcorrespondence for a scalar field coupled to 3d spin-3 gravity is realized inthis auxiliary 8d space. A bulk-to-boundary propagator of a scalar field isfound and a generating functional of boundary two-point functions of scalar${\cal W}$-descendant operators is obtained by using the classical action forthe scalar field. Classically, the scalar field must satisfy both Klein-Gordonequation and a third-order differential equation, which are related to thequadratic and cubic Casimir operators of $su(1,2)$. It is found that thecoefficient function of the derivatives of the scalar field in the latterequation is the spin-3 gauge field, when restricted to the hypersurface. Anaction integral in the 8d auxiliary space for the 3d spin-3 gravity coupled toa scalar field is presented. An 8d local frame is introduced and the equationsof motion for the 8d connections $A_{\mu}$, $\overline{A}_{\mu}$ are solved. Byrestricting those solutions onto $\Sigma$, flat connections in 3d$SL(3,\mathbb{R}) \times SL(3,\mathbb{R})$ Chern-Simons theory are obtained andnew 3d black hole solutions with and without spin-3 charge are found by thismethod.

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