Observe that ${{\chi }_{\mu \nu }}^{\rho \sigma }$ can be compactly expressed as ${{\chi }_{\mu \nu }}^{\rho \sigma }={{\mathrm{\Theta }}_{\mu \nu }}^{\rho \sigma }+\sqrt{{{\delta }_{[\mu }}^{[\rho }{{\delta }_{\nu ]}}^{\sigma ]}+{{{\mathrm{\Theta }}^2}_{\mu \nu }}^{\rho \sigma }}$, as in [53].

Expressed in this way, it might appear that the equations of motion are of third order in derivatives. However, they can be seen to be equivalent to the (second-order) equations ${\nabla }_{\mu }({\chi }^{\mu \nu \rho \sigma }{F}_{\rho \sigma })=0$ in the gauge $A_u=0$ by appropriate manipulations which allow to integrate them into second-order equations, uniquely fixed after requiring $A_{\mu }=0$ to be a solution.

Boundary indices are lowered and raised with the Minkowski metric ${\eta }^{ab}$.

The match is up to a constant on the right-hand side of (18), which corresponds to the different conventions used for the electromagnetic duality transformation.

We have chosen the sign for $K^L(\omega ,0)=K^T(\omega ,0)$ that guarantees a positive spectral function, defined by $-2\mathrm{Im}(\sqrt{k^2-{\omega }^2}{K}^T)$. The same result could also be derived by direct resolution of (10) for $k=0$.

This hydrodynamic-to-collisionless crossover manifests in the longitudinal conductivity and not in the transverse one since the longitudinal correlator $C_{xx}$ is related via Eq. (15) to the density-density correlator $C_{tt}$, which is a clear probe of hydrodynamic behavior [58].

If one insists on working with the GR solution, the lowest-order choice for ${\mathcal{T}}_{\mu \nu }$ which is nonzero on the AdS black brane solution of GR is the traceless part of ${\nabla }_{\mu }{W}_{\alpha \beta \rho \sigma }{\nabla }_{\nu }{W}^{\alpha \beta \rho \sigma }$, where $W_{\alpha \beta \rho \sigma }$ stands for the Weyl tensor. This term would in principle modify the conductivities of the dual CFT. Nevertheless, such a term is of order $\mathcal{O}(L^8)$ and there are other higher-curvature terms of less order that already modify the gravitational background.