Hydrodynamic Fluctuations, Long-time Tails, and Supersymmetry

TL;DR

The paper develops a general framework for hydrodynamic fluctuations at finite temperature and shows that long-time power-law tails, t^{-3/2}, arise from nonlinear couplings to conserved modes. Extending to supersymmetric theories, it introduces Grassmann-valued supercharge densities and derives supersymmetric constitutive relations, revealing that SUSY does not eliminate tails, but introduces a 1/N_c^2 suppression in their amplitude. Applied to N=4 SYM via AdS/CFT, the authors provide explicit predictions for the stress-tensor tail and its non-analytic small-frequency behavior, with loop (1/N_c^2) corrections required on the gravity side to reproduce these effects. The results offer a concrete, testable link between finite-temperature hydrodynamics and holographic duals, and suggest that long-time tails can probe the AdS/CFT correspondence through subleading, loop-level gravitational fluctuations.

Abstract

Hydrodynamic fluctuations at non-zero temperature can cause slow relaxation toward equilibrium even in observables which are not locally conserved. A classic example is the stress-stress correlator in a normal fluid, which, at zero wavenumber, behaves at large times as t^{-3/2}. A novel feature of the effective theory of hydrodynamic fluctuations in supersymmetric theories is the presence of Grassmann-valued classical fields describing macroscopic supercharge density fluctuations. We show that hydrodynamic fluctuations in supersymmetric theories generate essentially the same long-time power-law tails in real-time correlation functions that are known in simple fluids. In particular, a t^{-3/2} long-time tail must exist in the stress-stress correlator of N=4 supersymmetric Yang-Mills theory at non-zero temperature, regardless of the value of the coupling. Consequently, this feature of finite-temperature dynamics can provide an interesting test of the AdS/CFT correspondence. However, the coefficient of this long-time tail is suppressed by a factor of 1/N_c^2. On the gravitational side, this implies that these long-time tails are not present in the classical supergravity limit; they must instead be produced by one-loop gravitational fluctuations.