On the temperature dependence of the shear viscosity and holography

We examine the structure of the shear viscosity to entropy density ratio η/s in holographic theories of gravity coupled to a scalar field, in the presence of higher derivative corrections. Thanks to a non-trivial scalar field profile, η/s in this setup generically runs as a function of temperature. In particular, its temperature behavior is dictated by the shape of the scalar potential and of the scalar couplings to the higher derivative terms. We consider a number of dilatonic setups, but focus mostly on phenomenological models that are QCD-like. We determine the geometric conditions needed to identify local and global minima for η/s as a function of temperature, which translate to restrictions on the signs and ranges of the higher derivative couplings. Finally, such restrictions lead to an holographic argument for the existence of a global minimum for η/s in these models, at or above the deconfinement transition.