Second order equilibrium transport in strongly coupled $\mathcal{N} = 4$ supersymmetric $SU(N_c)$ Yang-Mills plasma via holography

Abstract: A relativistic fluid in 3+1 dimensions with a global $U(1)$ symmetry admits nine independent static susceptibilities at the second order in the hydrodynamic derivative expansion, which capture the response of the fluid in thermal equilibrium to the presence of external time-independent sources. Of these, seven are time-reversal $\mathbb{T}$ invariant and can be obtained from Kubo formulas involving equilibrium two-point functions of the energy-momentum tensor and the $U(1)$ current. Making use of the gauge/gravity duality along with the aforementioned Kubo formulas, we compute all seven $\mathbb{T}$ invariant second order susceptibilities for the $\mathcal{N} = 4$ supersymmetric $SU(N_c)$ Yang-Mills plasma in the limit of large $N_c$ and at strong 't-Hooft coupling $\lambda$. In particular, we consider the plasma to be charged under a $U(1)$ subgroup of the global $SU(4)$ R-symmetry of the theory. We present analytic expressions for three of the seven $\mathbb{T}$ invariant susceptibilities, while the remaining four are computed numerically. The dual gravitational description for the charged plasma in thermal equilibrium in the absence of background electric and magnetic fields is provided by the asymptotically AdS$_5$ Reissner-Nordstr\"{o}m black brane geometry. The susceptibilities are extracted by studying perturbations to the bulk geometry as well as to the bulk gauge field. We also present an estimate of the second order transport coefficient $\kappa$, which determines the response of the fluid to the presence of background curvature, for QCD, and compare it with previous determinations made using different techniques.