Higher spin contributions to holographic fluid dynamics inAdS5/CFT4

Abstract

We calculate the graviton's $\ensuremath{\beta}$ 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 ${\ensuremath{\beta}}_{mn}={R}_{mn}+4{T}_{mn}(g,u)$ (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 $T=\frac{1}{\ensuremath{\pi}}$. The stress-energy tensor is given by ${T}_{mn}={g}_{mn}+4{u}_{m}{u}_{n}+\ensuremath{\sum}_{N}{T}_{mn}^{(N)}$ where $u$ is the vector excitation satisfying ${u}^{2}=\ensuremath{-}1$ 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 $\ensuremath{\beta}$ 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 ${\mathrm{AdS}}_{5}$.