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Holographic parity violating charged fluid dual to Chern-Simons modified gravity

We discuss the $(2+1)$-dimensional parity violating charged fluid on a finite cutoff surface $Σ_c$, dual to the nondynamical and dynamical Chern-Simons (CS) modified gravities. Using nonrelativistic long-wavelength expansion method, the field equations are solved up to $\mathcal{O}(ε^2)$ in the nondynamical model. It is shown that there exists nonvortical dual fluid with shear viscosity $η$ and Hall viscosity $η_A$ on the cutoff surface $Σ_c$. The ratio of shear viscosity over entropy density $η/s$ of the fluid takes the universal value $1/{4π}$, while the ratio of Hall viscosity over entropy density $η_A/s$ depends on the $Σ_c$ and black brane charge $q$. Moreover the nonvortical dual fluid obeys the magnetohydrodynamic (MHD) equation. However, these kinematic viscosities $ν$ and $ν_A$ related to $η$ and $η_A$ do not appear in this MHD equation, due to the constraint condition $\tilde{\partial}^2β_j=0$ for the $(2+1)$-dimensional dual fluid. Then, we extend our discussion to the dynamical CS modified gravity and show that the dual vortical fluid possesses another so-called Curl viscosity $ζ_A$, whose ratio to entropy density $ζ_A/s$ also depends on the $Σ_c$ and $q$. Moreover, the value of $η/s$ still equals to $1/4π$ and the result of $η_A/s$ agrees to the previous result under the probe limit of the pseudo scalar field at the infinite boundary in the charged black brane background for the dynamical CS modified gravity. This vortical dual fluid corresponds to the magnetohydrodynamic (MHD) turbulence equation in plasma physics.