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Higher Spin Contributions to Holographic Fluid Dynamics in $AdS_5/CFT_4$

We calculate the graviton's $β$-function in $AdS$ string-theoretic sigma-model, perturbed by vertex operators for Vasiliev's higher spin gauge fields in $AdS_5$. The result is given by $β_{mn}=R_{mn}+4T_{mn}(g,u)$ (with the AdS radius set to 1 and the graviton polarized along the $AdS_5$ boundary), with the matter stress-energy tensor given by that of conformal holographic fluid in $d=4$, evaluated at the temperature given by $T={1\overπ}$. The stress-energy tensor is given by $T_{mn}={g_{mn}}+4u_mu_n+\sum_{N}T^{(N)}_{mn}$ where $u$ is the vector excitation satisfying $u^2=-1$ and $N$ is the order of the gradient expansion in the dissipative part of the tensor. We calculate the contributions up to N=2. The higher spin excitations contribute to the $β$-function, ensuring the overall Weyl covariance of the matter stress tensor. We conjecture that the structure of gradient expansion in $d=4$ conformal hydrodynamics at higher orders is controlled by the higher spin operator algebra in $AdS_5$.