Higher spin contributions to holographic fluid dynamics in ${\mathrm{AdS}}_5/{\mathrm{CFT}}_4$

We calculate the graviton’s $\beta$ function in the AdS string-theoretic sigma model, perturbed by vertex operators for Vasiliev’s higher spin gauge fields in ${\mathrm{AdS}}_5$. The result is given by ${\beta }_{mn}=R_{mn}+4T_{mn}(g,u)$ (with the AdS radius set to 1 and the graviton polarized along the ${\mathrm{AdS}}_5$ boundary), with the matter stress-energy tensor given by that of conformal holographic fluid in $d=4$, evaluated at the temperature given by $T=\frac{1}{\pi }$. The stress-energy tensor is given by $T_{mn}=g_{mn}+4u_m{u}_n+\sum _N{T}_{mn}^{(N)}$ 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 $\beta$ 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 ${\mathrm{AdS}}_5$.