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

Abstract

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 $W_n(n=0,\pm 1,\pm 2)$ and $L_n(n=0,\pm 1)$ 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 Poincaré 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 $e^{i{x}^{+}{L}_{-1}^h}{e}^{{\beta }^{+}{W}_{-1}^h}{\Phi }_h(0)e^{-{\beta }^{+}{W}_{-1}^h}{e}^{-i{x}^{+}{L}_{-1}^h}$. Here $L_n^h$ and $W_n^h$ are sl(3,R) generators in the hyperbolic basis for Poincaré coordinates. It is shown that in the ${\beta }^{+}\to \infty$ limit, the conformal weight changes to a new value $h^{\prime }=h/2$. This may be regarded as a Renormalization Group (RG) flow. It is argued that this RG flow will be triggered by terms $\Delta S\propto {\beta }^{+}{W}_{-1}^h+{\beta }^{-}{\overline{W}}_{-1}^h$ added to the action.

arXiv comment27pp. no figures

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Cited by in corpus (2)

• Wilson Lines and Ishibashi States in AdS${}_3$/CFT${}_2$
• Holographic Duality for 3D Spin-3 Gravity Coupled to Scalar Field