One-loop corrected holographic shear viscosity to entropy density ratio at low temperatures

A

bstract

Near-extremal black holes contain infrared-enhanced quantum fluctuations in their near-horizon near-AdS ₂ throat, governed in part by Schwarzian modes. We study the effect of these fluctuations on the zero-frequency stress-tensor response of a near-extremal asymptotically AdS ₄ Reissner-Nordström black brane whose transverse directions are regulated by a finite toroidal quotient. Working directly in four dimensions, we compute the leading correction in the weakly coupled Schwarzian regime T _q  ≪  T  ≪  r ₀ / L ² . At one loop, the shear perturbation couples to the n = 2 would-be zero mode and produces a correction proportional to T _q / T . Together with the logarithmic correction to the entropy, this yields a finite-volume correction to η / s . Our calculation provides a direct four-dimensional benchmark, complementary to exact two-dimensional treatments, while keeping the asymptotic AdS ₄ source explicit.