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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., Comment: 44 pages, 1 figure

We construct a localized state of a scalar field in 3D spin-3 gravity. 3D spin-3 gravity is thought to be holographically dual to W$_3$ extended CFT on a boundary at infinity. It is known that while W$_3$ algebra is a non-linear algebra, in the limit of large central charge $c$ a linear finite-dimensional subalgebra generated by $W_n \, (n=0,\pm1,\pm2)$ and $L_n (n= 0,\pm1)$ is singled out. The localized state is constructed in terms of these generators. To write down an equation of motion for a scalar field which is satisfied by this localized state it is necessary to introduce new variables for an internal space $\alpha^{\pm}$, $\beta^{\pm}$, $\gamma$, in addition to ordinary coordinates $x^{\pm}$ and $y$. The higher-dimensional space, which combines the bulk spacetime with the `internal space', which is an analog of superspace in supersymmetric theory, is introduced. The `physical bulk spacetime' is a 3D hypersurface with constant $\alpha^{\pm}$, $\beta^{\pm}$ and $\gamma$ embedded in this space. We will work in Poincar\'e coordinates of AdS space and consider W-quasi-primary operators $\Phi_{h}(x^+)$ with a conformal weight $h$ in the boundary and study two and three point functions of W-quasi-primary operators transformed as $e^{ix^+L^h_{-1}} e^{\beta^+W^h_{-1}} \Phi_{h}(0) e^{-\beta^+W^h_{-1}}e^{-ix^+L^h_{-1}}$. Here $L^h_n$ and $W^h_n$ are sl(3,R) generators in the hyperbolic basis for Poincar\'e coordinates. It is shown that in the $\beta^+ \rightarrow \infty$ limit, the conformal weight changes to a new value $h'=h/2$. This may be regarded as a Renormalization Group (RG) flow. It is argued that this RG flow will be triggered by terms $\Delta S \propto \beta^+ W^h_{-1}+\beta^- \overline{W}^h_{-1}$ added to the action., Comment: 27pp. no figures

In this study, a method to construct a full-colour volumetric display is presented using a commercially available inkjet printer. Photoreactive luminescence materials are minutely and automatically printed as the volume elements, and volumetric displays are constructed with high resolution using easy-to-fabricate means that exploit inkjet printing technologies. The results experimentally demonstrate the first prototype of an inkjet printing-based volumetric display composed of multiple layers of transparent films that yield a full-colour three-dimensional (3D) image. Moreover, we propose a design algorithm with 3D structures that provide multiple different 2D full-colour patterns when viewed from different directions and experimentally demonstrates prototypes. It is considered that these types of 3D volumetric structures and their fabrication methods based on widely deployed existing printing technologies can be utilised as novel information display devices and systems, including digital signage, media art, entertainment and security.

In this paper we will consider the Almheiri-Polchinski model of the AdS$_2$ back reaction coupled with Liouville field, which is necessary for quantum consistency. In this model, the Liouville field is determined classically by a bulk conformal transformation. The boundary time is also reparametrized by this transformation. It is shown that the on-shell action on the boundary for the gravity sector is given by a bulk integral containing the Liouville field. This integral stems from Weyl anomaly and is SL(2,R) invariant. A prescription is given for computing correlation functions of the operators dual to massless scalars. The generating function of the correlation functions of these operators is given by a sum of matter action and the bulk integral containing the Liouville field. The latter integral leads to extra contributions to $n(\geq 6)$ point functions., Comment: 18 pages