On the universal identity in second order hydrodynamics

TL;DR

This work tests the robustness of the second-order hydrodynamic identity $H=2\eta\tau_{\Pi}-4\lambda_1-\lambda_2$ in strongly coupled conformal theories with gravity duals. It computes the $\mathcal{O}(\lambda^{-3/2})$ correction to the second-order coefficient $\lambda_2$ in ${\cal N}=4$ SYM from $R^4$ terms in type IIB string theory and shows that the Haack–Yarom relation $H=0$ remains valid at this order. It further extends the analysis to linear curvature-squared corrections (including Gauss–Bonnet gravity) and confirms that $H$ is preserved to linear order in the higher-derivative couplings, providing a broader sense in which these near-equilibrium dissipation relations are constrained at strong coupling. The results employ holographic three-point stress-energy tensor correlators and field-redefinition techniques to connect higher-derivative gravity to effective two-derivative descriptions, with implications for entropy production in near-equilibrium conformal fluids.

Abstract

We compute the 't Hooft coupling correction to the infinite coupling expression for the second order transport coefficient $λ_2$ in ${\cal N}=4$ $SU(N_c)$ supersymmetric Yang-Mills theory at finite temperature in the limit of infinite $N_c$, which originates from the $R^4$ terms in the low energy effective action of the dual type IIB string theory. Using this result, we show that the identity involving the three second order transport coefficients, $2 ητ_Π- 4 λ_1 - λ_2 =0$, previously shown by Haack and Yarom to hold universally in relativistic conformal field theories with string dual descriptions to leading order in supergravity approximation, holds also at next to leading order in this theory. We also compute corrections to transport coefficients in a (hypothetical) strongly interacting conformal fluid arising from the generic curvature squared terms in the corresponding dual gravity action (in particular, Gauss-Bonnet action), and show that the identity holds to linear order in the higher-derivative couplings. We discuss potential implications of these results for the near-equilibrium entropy production rate at strong coupling.