One-loop Corrected Holographic Shear Viscosity to Entropy Density Ratio at Low Temperatures

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 squared. 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.