Universal thermal and electrical conductivity from holography

Abstract

It is known from earlier work of Iqbal, Liu [1] that the boundary transport coefficients such as electrical conductivity (at vanishing chemical potential), shear viscosity etc. at low frequency and finite temperature can be expressed in terms of geometrical quantities evaluated at the horizon. In the case of electrical conductivity, at zero chemical potential gauge field fluctuation and metric fluctuation decouples, resulting in a trivial flow from horizon to boundary. In the presence of chemical potential, the story becomes complicated due to the fact that gauge field and metric fluctuation can no longer be decoupled. This results in a nontrivial flow from horizon to boundary. Though horizon conductivity can be expressed in terms of geometrical quantities evaluated at the horizon, there exist no such neat result for electrical conductivity at the boundary. In this paper we propose an expression for boundary conductivity expressed in terms of geometrical quantities evaluated at the horizon and thermodynamic quantities. We also consider the theory at finite cutoff recently constructed in [2], at radius r _c outside the horizon and give an expression for cutoff dependent electrical conductivity (σ(r _c )), which interpolates smoothly between horizon conductivity σ _H (r _c → r _h ) and boundary conductivity σ _B (r _c → ∞). Using the results about the conductivity we gain much insight into the universality of thermal conductivity to viscosity ratio proposed in [3].