Holography of charged dilaton black holes

We study charged dilaton black branes in AdS4. Our system involves a dilaton ϕ coupled to a Maxwell field Fμν with dilaton-dependent gauge coupling, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \frac{1}{{{g^2}}} = {f^2}\left( \phi \right) $\end{document}. First, we find the solutions for extremal and near extremal branes through a combination of analytical and numerical techniques. The near horizon geometries in the simplest cases, where f(ϕ) = eαϕ, are Lifshitz-like, with a dynamical exponent z determined by α. The black hole thermodynamics varies in an interesting way with α, but in all cases the entropy is vanishing and the specific heat is positive for the near extremal solutions. We then compute conductivity in these backgrounds. We find that somewhat surprisingly, the AC conductivity vanishes like ω2 at T = 0 independent of α. We also explore the charged black brane physics of several other classes of gauge-coupling functions f(ϕ). In addition to possible applications in AdS/CMT, the extremal black branes are of interest from the point of view of the attractor mechanism. The near horizon geometries for these branes are universal, independent of the asymptotic values of the moduli, and describe generic classes of endpoints for attractor flows which are different from AdS2 × R2.