Coupling constant dependence of the shear viscosity in N=4 supersymmetric Yang-Mills theory

Gauge theory - gravity duality predicts that the shear viscosity of N=4 supersymmetric SU(N_c) Yang-Mills plasma at temperature T in the limit of large N_c and large 't Hooft coupling g^2_{YM} N_c is independent of the coupling and equals to \pi N_c^2 T^3/8. In this paper, we compute the leading correction to the shear viscosity in inverse powers of 't Hooft coupling using the \alpha'-corrected low-energy effective action of type IIB string theory. We also find the correction to the ratio of shear viscosity to the volume entropy density (equal to 1/4\pi in the limit of infinite coupling). The correction to 1/4\pi scales as (g^2_{YM} N_c)^{-3/2} with a positive coefficient.