A Bulk Localized State and New Holographic Renormalization Group Flow in 3D Spin-3 Gravity

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.