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AdS/CFT for 3D Higher-Spin Gravity Coupled to Matter Fields

New holographic prescription for the model of 3d higher-spin gravity coupled to real matter fields $B_{μν}$ and $C$, which was introduced in ArXiv:1304.7941[hep-th], is formulated. By using a local symmetry, two of the components of $B_{μν}$ are eliminated, and gauge-fixing conditions are imposed such that the non-vanishing component, $B_{ϕρ}$, satisfies a covariantly-constancy condition in the background of Chern-Simons gauge fields $A_μ$, $\bar{A}_μ$. In this model, solutions to the classical equations of motion for $A_μ$ and $\bar{A}_μ$ are non-flat due to the interactions with matter fields. The solutions for the gauge fields can, however, be split into two parts, flat gauge fields ${\cal A}_μ$, $\bar{\cal A}_μ$, and those terms that depend on the matter fields. The equations for the matter fields then coincide with covariantly-constancy equations in the flat backgrounds ${\cal A}_μ$ and $\bar{\cal A}_μ$, which are exactly the same as those in linearized 3d Vasiliev gravity. The two- and three-point correlation functions of operators in the boundary CFT are computed by using an on-shell action, $ \text{tr}\, (B_{ϕρ} \, C)$. This term does not depend on coordinates due to the matter equations of motion, and it is not necessary to take the near-boundary limit $ρ\rightarrow \infty$. Analysis is presented for SL(3,R) $\times$ SL(3,R) as well as $HS[\frac{1}{2}] \times HS[\frac{1}{2}]$ higher-spin gravity. In the latter model, scalar operators with scaling dimensions $Δ_+=\frac{3}{2}$ and $Δ_-=\frac{1}{2}$ appear in a single quantization.