Black holes and non-relativistic quantum systems

We describe black holes in d+3 dimensions, whose thermodynamic properties correspond to those of a scale invariant non-relativistic d+1 dimensional quantum system with dynamical exponent z=2. The gravitational model involves a massive abelian vector field and a scalar field, in addition to the metric. The energy per particle in the dual theory is $\mathrm{\mid }\mu \mathrm{\mid }d\mathrm{/}(d+2)$, exactly as in a non-interacting Fermi gas, while the ratio of shear viscosity to entropy density is $\mathrm{\hslash }\mathrm{/}4\pi$.