Shear Viscosity of Strongly Coupled $N=4\mathrm{}$ Supersymmetric Yang-Mills Plasma

Using the anti–de Sitter/conformal field theory correspondence, we relate the shear viscosity $\eta$ of the finite-temperature $N=4\mathrm{}$ supersymmetric Yang-Mills theory in the large $N$, strong-coupling regime with the absorption cross section of low-energy gravitons by a near-extremal black three-brane. We show that in the limit of zero frequency this cross section coincides with the area of the horizon. From this result we find $\eta =\frac{\pi }{8}{N}^2{T}^3$. We conjecture that for finite ’t Hooft coupling $g_{\mathrm{YM}}^{2}N$ the shear viscosity is $\eta =f\left(g_{\mathrm{YM}}^{2}N\right)N^2{T}^3$, where $f\left(x\right)$ is a monotonic function that decreases from $O\left(x^{-2\mathrm{}}{ln}^{-1\mathrm{}}\right(1/x\left)\right)$ at small $x$ to $\pi /8\mathrm{}$ when $x\to \infty$.